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<h1>The Flaw in Gödel’s proof of his Incompleteness theorem</h1>
<p class="indent2 font80">Page last updated 18 Oct 2025</p>
<p class="line3">&nbsp;</p>
<p>I had previously considered that it would not be feasible to give a brief summary of the flaw in the proof of <a class="tt_incomp tt" href="/topics/formal_language#incomp">incompleteness</a> (which you can see online at <a href="/ffgit/godel-original-english">English translation of Gödel’s original proof</a>) by Kurt Gödel, a proponent of intelligent design (see <a href="/ffgit/godel-quotes">Statements by Kurt Gödel</a>). But in fact it is possible to pinpoint an intuitive assumption that Gödel makes in his proof which is easily demonstrated to be incorrect and logically invalid. Since the remainder of his proof relies completely on this assumption, this renders the result of the proof invalid.</p>
<p>&nbsp;</p>
<p>That assumptive assertion is what Gödel calls Relation 17, and which states:</p>
<p class="indent2"><eqn-span>Z(<i>n</i>) = <i>n</i>&#8239;N&#8239;[R(1)]</eqn-span></p>
<p class="indent2"><eqn-span>Z(<i>n</i>)</eqn-span> is the <i class="bluebg">number-string</i> for the number&nbsp;<eqn-i>n</eqn-i>.⁠<span class="footnote">
	“Z(<i>n</i>) ist das Zahlzeichen für die Zahl <i>n”</i> in the original German. The place in the proof where this occurs can be seen in the online English translation at <a href="/ffgit/godel-original-english#REL-17">Gödel’s Proof: Relation 17</a>.</span></p>
<p>&nbsp;</p>
<p>Gödel previously defined that an italicized word, such as <i class="bluebg">number-string</i> indicates the number calculated by his Gödel numbering function <eqn-span>Φ</eqn-span>, when applied to a given sequence of symbols of the formal system&nbsp;P (the system for which Gödel is claiming incompleteness). In other words, he is claiming:</p>
<p class="indent2 spacing2"><eqn-span>Z(<i>n</i>) = Φ(<i>n</i>)</eqn-span></p>
<p>such that their calculated values are equal when their free <a class="tt_variable tt" href ="/ffgit/GodelSimplified1#variable">variables</a> are equal, and the result of his paper has a complete reliance on this assertion, which is an assumption by Gödel for which he offers no logical justification whatsoever.</p>
<p>&nbsp;</p>
<p>It is easily shown that no such equivalence can apply, as follows:</p>
<p>&nbsp;</p>
<div class="indent2">
<p>The <a class="tt_domain tt" href="/ffgit/errors_inc_proofs#domain">domain</a> of the free variable&nbsp;<eqn-i>n</eqn-i> of&nbsp;<eqn-span>Z</eqn-span> is defined to be all <a class="tt_natural tt" href="/topics/glossary#natural">natural numbers</a>, i.e: the only values can be substituted for the&nbsp;<eqn-i>n</eqn-i> are the natural numbers <eqn-span>0, 1, 2, 3, … </eqn-span>. On the other hand, the domain of the free variable of <eqn-span>Φ</eqn-span> is all sequences of symbols of the formal system&nbsp;P, an example being</p>
<p><eqn-span><i>x</i><sub>2</sub>(0) ∧</eqn-span> <eqn-span><i>x</i><sub>1</sub>∀&#8239;(<i>x</i><sub>2</sub>(<i>x</i><sub>1</sub>) ⊃ <i>x</i><sub>2</sub>(<i>&#8202;f&#8239;x</i><sub>1</sub>)) ⊃</eqn-span> <eqn-span><i>x</i><sub>1</sub><span class="letterSp-1">∀&#8239;</span>(<i>x</i><sub>2</sub>(<i>x</i><sub>1</sub>))</eqn-span>.</p>
<p>&nbsp;</p>
<p>Clearly Gödel did not intend the complete domain of <eqn-span>Φ</eqn-span> to be all sequences of the formal system&nbsp;P in the context of this assumed equivalence to the <eqn-i>Z</eqn-i> function, but we can restrict the domain of&nbsp;<eqn-i>n</eqn-i> in both <eqn-span>Z(<i>n</i>)</eqn-span> and <eqn-span>Φ(<i>n</i>)</eqn-span> to be only natural numbers, where the format used for natural numbers in both the formal system and for the&nbsp;<eqn-span>Z</eqn-span> function is to be in the form of <eqn-span>0, <i>f</i>&#8239;0, <i>ff</i>&#8239;0, <i>fff</i>&#8239;0 &hellip; </eqn-span>, (the format for natural numbers in the formal system) so that the formats of the numbers that constitute the domain of the free variables of the&nbsp;<eqn-span>Z</eqn-span> function and his numbering function <eqn-span>Φ</eqn-span> are precisely identical.</p>
<p>&nbsp;</p>
<p>But we don’t even need that restriction - we can observe that Gödel’s paper can be rewritten where numbers are to be in the form of <eqn-span><i>fff</i>&hellip;0</eqn-span> rather than conventional decimal format. This does not change any meaning or result of the proof, but it means that all numbers in the proof are automatically in the form of <eqn-span><i>fff</i>&hellip;0</eqn-span>.</p>
<p>&nbsp;</p>
<p>However, this still does not achieve an equivalence of the two functions, and nor does a restriction on the domain of the number <eqn-span><i>n</i></eqn-span>. This is because there can be infinitely many expressions that have the same value as any given number, and which are also valid values for the substitution of the free variable of the&nbsp;<eqn-span>Z</eqn-span> function. For example, in the format of the formal system, 3 is <eqn-span><i>fff</i>&#8239;0</eqn-span>, 2 is <eqn-span><i>ff</i>&#8239;0</eqn-span> and&nbsp;5 is <eqn-span><i>fffff</i>&#8239;0</eqn-span>, so we have that <eqn-span><i>fff</i>&#8239;0 + <i>ff</i>&#8239;0 = <i>fffff</i>&#8239;0</eqn-span> and both <eqn-span><i>fff</i>&#8239;0 + <i>ff</i>&#8239;0</eqn-span> and <eqn-span><i>fffff</i>&#8239;0</eqn-span> are precisely the same value when substituted for the free variable of&nbsp;<eqn-span>Z</eqn-span>, and the&nbsp;<eqn-span>Z</eqn-span> function gives precisely the same value for either of these two substitutions.</p>
<p>&nbsp;</p>
<p>But the <i><b>sequences of symbols</b></i> “<eqn-span><i>fff</i>&#8239;0 + <i>ff</i>&#8239;0</eqn-span>” and “<eqn-span><i>fffff</i>&#8239;0</eqn-span>” are <i><b>not</b></i> the same value when substituted for the free variable of <eqn-span>Φ</eqn-span> - the <eqn-span>Φ</eqn-span> function, by definition, must <i><b>always</b></i> give a different value for <i><b>every</b></i> different sequence of symbols of the formal system, regardless of any calculated value of that sequence of symbols. For the substitution of “<eqn-span><i>fffff</i>&#8239;0</eqn-span>” ,the <eqn-span>Φ</eqn-span> function gives the result:</p>
<p class="indent2 spacing2"><eqn-span>2<sup>3</sup>&nbsp;·&nbsp;3<sup>3</sup>&nbsp;·&nbsp;5<sup>3</sup>&nbsp;·&nbsp;7<sup>3</sup>&nbsp;·&nbsp;11<sup>3</sup>&nbsp;·&nbsp;13<sup>1</sup></eqn-span></p>
<p>i.e, a result with 6 prime factors, while even with a formal system that includes the “+” symbol, for the substitution of “<eqn-span><i>fff</i>&#8239;0 + <i>ff</i>&#8239;0</eqn-span>” the <eqn-span>Φ</eqn-span> function must give a result that has at least 8 prime factors, viz:⁠<span class="footnote">
	Note: There is no “+” symbol in Gödel’s formal system&nbsp;P, and the concept of addition in that system requires the use of many more symbols, so the result would have many more than 8 prime factors.</span></p>
<p class="indent2 spacing2"><eqn-span>2<sup>3</sup>&nbsp;·&nbsp;3<sup>3</sup>&nbsp;·&nbsp;5<sup>3</sup>&nbsp;·&nbsp;7<sup>1</sup>&nbsp;·&nbsp;11<sup><i>x</i></sup>&nbsp;·&nbsp;13<sup>3</sup>&nbsp;·&nbsp;17<sup>3</sup>&nbsp;·&nbsp;19<sup>1</sup></eqn-span></p>
<p>Hence it is quite obvious that there cannot be an equivalence of the <eqn-span>Z(<i>n</i>)</eqn-span> and <eqn-span>Φ(<i>n</i>)</eqn-span> functions as Gödel intuitively and incorrectly assumes, since they can result in different values for equal valued substitutions of their free variables.</p>
</div>
<p>&nbsp;</p>
<p>Gödel uses that incorrect assumption of equivalence as an essential part of his proof of his assertion that he calls Proposition&nbsp;V, which renders his Proposition&nbsp;V has logically invalid and the remainder of his proof that follows is likewise completely invalid. See also <a href="/ffgit/godel-guide-8#Zfunction">The Z function assumption</a>.</p>
<p>&nbsp;</p>
<p>Perhaps the reader might think that perhaps the above demonstrated error of assumed equivalence is only a curiosity, and perhaps there is some way to circumvent it. But there isn’t. Several people have come up with bizarre ideas in attempts to circumvent the flaw, on web-pages such as on <a href="https://www.reddit.com/r/badmathematics/comments/1kxz5ou/the_fundamental_flaw_in_g%C3%B6dels_proof_of_the/" target="_blank" rel="noopener noreferrer">Reddit - Gödel’s flaw</a>, where rhetoric, denial of facts and cherry-picking get more upvotes than scholarly logical argument - I detail some of these in the following section.</p>
<p class="line7">&nbsp;</p>
<div class="appendixQ bg1X cmtBox" id="WeirdIdeas">
<h3 class="marginTop5">Attempts to evade the contradiction</h3>
<p>Various people have come up with weird notions in attempts to evade the inherent flaw in Gödel’s proof. In this section I point out the fallacies in some of these bizarre attempts to rescue Gödel’s proof:</p>
<p class="line7">&nbsp;</p>

<h5>Weird idea 1: Gödel didn’t intend an equivalence of his <eqn-span>Z</eqn-span> function and the Gödel numbering function</h5>
<p>This assertion claims that Gödel did not really mean what he actually wrote in his <a href="/ffgit/godel-original-english#REL-17">Relation 17</a>:</p>
<p class="indent"><eqn-span>Z(<i>n</i>) ≡ <i>n</i> N [R(1)]</eqn-span></p>
<p class="indent"><eqn-span>Z(<i>n</i>)</eqn-span> is the <i class="bluebg">number-string</i> for the number <eqn-i>n</eqn-i>.</p>
<p class="line5">&nbsp;</p>
<p>where Gödel had previously defined that words in italics indicate numbers that are Gödel numbers of the formal sequence that is described by that same word without italics, and a number-string is a sequence of the form <eqn-span><i>fff</i>&hellip;0</eqn-span>, i.e: <eqn-span>Z(<i>n</i>)</eqn-span> is the Gödel number of <eqn-i>n</eqn-i>, in other words:</p>
<p class="indent"><eqn-span>Z(<i>n</i>) = Φ(<i>n</i>)</eqn-span></p>
<p>where <eqn-span>Φ</eqn-span> is the Gödel numbering function, where Gödel is clearly assuming that sequences of the form <eqn-span><i>fff</i>&hellip;0</eqn-span> are also valid as numbers for number-theoretic relations.</p>
<p class="line5">&nbsp;</p>
<p>It’s not as if there’s any room for nuance in Gödel’s statement in his original German:</p>
<p class="indent"><eqn-span>Z(<i>n</i>)</eqn-span> ist das <i>Zahlzeichen</i> für die Zahl <eqn-i>n</eqn-i></p>
<p>and a straightforward word for word translation gives:</p>
<p class="indent"><eqn-span>Z(<i>n</i>)</eqn-span> is the <i>number-string</i> for the number <eqn-i>n</eqn-i></p>
<p>which is why all translations of this statement are essentially the same.&nbsp;Furthermore, when van Heijenoort was making a translation of Gödel’s paper, there was frequent communication between them
<span class="footnote">
	Over the period 1963-1965, see Kurt Gödel, <i>Collected Works</i> Volume V, Oxford University Press, 2013.</span>
since van Heijenoort wanted Gödel’s full approval of his translation before publication. In view of the fact that in that communication Gödel made numerous finely detailed suggestions, it’s clear that if Gödel had ever thought that there was anything wrong with the statement above regarding <eqn-span>Z(<i>n</i>)</eqn-span>, he had ample opportunities&nbsp; (over a period of over 30 years) to clarify what he had actually intended if it was not as he had written it in 1930.</p>
<p class="line12">&nbsp;</p>
<p>Moreover, further evidence that Gödel intended exactly what he wrote in his relation 17 - rather than making some sort of slip-up - can be seen by his use of the number-theoretic function <eqn-span>Sb&#8202;(<i>f &nbsp;<sup class="fract">v</sup>/<sub class="fract">y</sub></i>)</eqn-span> in Proposition&nbsp;5 of his paper. This function <eqn-span>Sb&#8202;(<i>f &nbsp;<sup class="fract">v</sup>/<sub class="fract">y</sub></i>)</eqn-span> is defined by Gödel (his <a href="/ffgit/godel-original-english#REL-31">relation 31</a>) to correspond to the substitution of the free variable&nbsp;<eqn-span>V</eqn-span> of the formal expression&nbsp;<eqn-span>F</eqn-span> by the formal expression&nbsp;<eqn-span>Y</eqn-span> where <eqn-span><i>f</i> = Φ(F)</eqn-span>, <eqn-span><i>y</i> = Φ(Y)</eqn-span>, and&nbsp;<eqn-i>v</eqn-i> corresponds to the formal variable&nbsp;<eqn-i>V</eqn-i>, where both  <eqn-i>f</eqn-i>  and  <eqn-i>y</eqn-i> must be Gödel numbers in order that the relationship between the formal expressions corresponds to the relationship between the corresponding Gödel numbers.</p>
<p>&nbsp;</p>
<p>And Gödel asserts in his <a href="/ffgit/godel-original-english#EQ3">equation 3</a> of his <a href="/ffgit/godel-original-english#PROP-V">Proposition&nbsp;5</a> that (note that here, for simplicity, the <eqn-span>Sb</eqn-span> function has only one free variable; this makes no difference to the principle involved):</p>
<div class="indent xScroll">
<p class="indent">(A) <eqn-span>R(<i>x</i>) ⇒ Bew{ Sb[<i>r</i>&nbsp;&nbsp;<sup class="fract"><i>u</i></sup> ⁄ <sub class="fract">Z(<i>x</i>) </sub>] } </eqn-span></p>
<p>where <eqn-span>Z(<i>x</i>)</eqn-span> takes the place of the third variable <eqn-i>y</eqn-i>, and which has to be a Gödel number to preserve the aforementioned correspondence between:</p>
<ul class="spacing1">
<li>relationships between formal expressions, and</li>
<li>relationships between the Gödel numbers of those expressions,</li>
</ul>
<p>so there is necessarily an assumption that <eqn-span>Z(<i>x</i>)</eqn-span> is a Gödel number, i.e: <eqn-span>Z(<i>n</i>) = Φ(<i>n</i>)</eqn-span>.</p>
</div>
<p class="line12">&nbsp;</p>
<p>And we note that an expression such as:</p>
<p class="indent xScroll">(B) <eqn-span>R(<i>x</i>) ⇒ Bew{ Sb[<i>r</i>&nbsp;&nbsp;<sup class="fract"><i>u</i></sup> ⁄ <sub class="fract">Φ(<i>x</i>) </sub> ] } </eqn-span></p>

<p>would be assuming that the <eqn-span>Φ</eqn-span> function is a number-theoretic function, which is clearly not the case. Instead, from the definition of the <eqn-span>Sb</eqn-span> function, it follows that equation (A) assumes the validity of:</p>
<div class="indent xScroll">
<p class="indent">(C) <eqn-span>R(<i>x</i>) ⇒ Bew{ Sb[<i>r&nbsp;&nbsp;<sup class="fract">u</sup> ⁄ <sub class="fract">w</sub></i> ] } </eqn-span></p>

	<p>where <eqn-span><i>w</i> = Z(W) = Φ(W)</eqn-span>,</p>
	<p>where <eqn-span>Φ</eqn-span> is the Gödel numbering function,</p>
	<p>and <eqn-span>W</eqn-span> is some expression that satisfies the domain of the free variables of both the <eqn-span>Z</eqn-span> function and the <eqn-span>Φ</eqn-span> function (i.e: with restrictions on the domain of <eqn-span>W</eqn-span>).</p>
</div>
<p class="line12">&nbsp;</p>
<p id="Footnote43a">On top of the above, in van Heijenoort’s approved translation, footnote 43 includes the parenthetical addition:</p>
<p class="indent">[Precisely stated the final part of this footnote (which refers to a side remark unnecessary for the proof) would read thus: "<span class="font80">REPLACEMENT</span> of a <span class="font80">VARIABLE</span> by the <span class="font80">NUMERAL</span> for p."].</p>
<p>This refers to Gödel’s equation 12, <eqn-span>r = Sb(q  <sup class="fract">19</sup>⁄<sub class="fract">Z(p)</sub> )</eqn-span>, where the variable <eqn-i>y</eqn-i> of the number-theoretic <eqn-span>Sb&#8202;(<i>x &nbsp;<sup class="fract">v</sup>/<sub class="fract">y</sub></i>)</eqn-span> has been substituted, not by <eqn-span>p</eqn-span>, but by <eqn-span>Z(p)</eqn-span>, thus confirming that, according to Gödel, <eqn-span>Z(p)</eqn-span> is precisely the same as the Gödel number of <eqn-span>p</eqn-span> - since in van Heijenoort’s translation the corresponding term for a number given by Gödel’s numbering function is indicated by small capitals, i.e the "<span class="font80">NUMERAL</span>" for <eqn-span>p</eqn-span> is the Gödel number of the “numeral” <eqn-span>p</eqn-span>, and in that translation a “numeral” is defined as a formal sequence with the format of 0, <i>f</i>0, <i>ff</i>0, <i>fff</i>0, &hellip; or <i>a</i>, <i>fa</i>, <i>ffa</i>, <i>fffa</i>, &hellip; . This also confirms that Gödel considered sequences so formatted as valid values for his number-theoretic relations.</p>
<p>&nbsp;</p>
<p>In short, all the evidence demonstrates that the claim that in reality Gödel did not intend what he had actually written is an absurd fantasy denial of the facts of the matter.<span class="footnote">
	One will also find instances of cherry-picking, for example, remarking on how hundreds of very clever perspicacious mathematicians have examined Gödel’s proof and none of them found anything wrong with it. What they omit to add is the concomitant fact that none of these very clever perspicacious mathematicians ever made any negative remark about Gödel’s assertion that the function
<eqn-span>Z(<i>n</i>)</eqn-span> is the Gödel number of <eqn-i>n</eqn-i> - which, according to the naysayers, is not actually literally correct.</span>
Gödel’s <a href="/ffgit/godel-original-english#PROP-V">Proposition&nbsp;5</a> assumes an equivalence of his <eqn-span>Z</eqn-span> function and the Gödel numbering function, over a restricted domain of the free variables, and as demonstrated above, that results in a contradiction.</p>
<p>&nbsp;</p>
<p>Uwe Petersen has published a critique of my work on Gödel’s incompleteness proof and which attempts to argue that no equivalence between the <eqn-span>Z</eqn-span> function and the Gödel numbering function is involved in the proof; my analysis of his paper is at <a href="/ffgit/uwe-petersen-on-godel-flaw">Uwe Petersen on Gödel’s incompleteness proof</a>.</p>
<p>&nbsp;</p>
<h6>Further weirdness</h6>
<p>Some people try claiming that Gödel could not have intended any sort of equivalence of the two functions on the grounds that the domain of the free variable of&nbsp;<eqn-span>Z</eqn-span> 	is objects that are natural numbers, while the domain of the Gödel numbering function is strings of symbols. But 57 is a string of two symbols and it is also a number. Every possible substitution of the variable of&nbsp;<eqn-span>Z</eqn-span> is a string of symbols (where a single symbol is a string of length one), and some of the strings of symbols of the formal system satisfy the definition of natural number, i.e: they satisfy the Peano axioms, hence, over a restricted domain the Gödel numbering function and the Z function both have a domain that satisfies the definition of natural numbers. And as noted above, Gödel approved a translation that includes a direct confirmation of the equivalence.</p>
<p>&nbsp;</p>
<p>Besides that, while Gödel’s paper is written with his number-theoretic relations and functions with natural numbers in conventional decimal format, there is no reason why one could not rewrite the paper using only the format of <eqn-span>0, <i>f</i>&#8239;0, <i>ff</i>&#8239;0, <i>fff</i>&#8239;0 &hellip; </eqn-span>for natural numbers, and the mathematical meaning and the result must remain unchanged, since the numerical values remain the same. And since that is the case, the notion that the weird appeal regarding the domains of &nbsp;<eqn-span>Z</eqn-span> and the Gödel numbering function can be seen to be nothing more than wishful thinking.</p>
<p>&nbsp;</p>
<h6>Even further weirdness</h6>
<p>In an attempt to show that Gödel did not really intend what he wrote regarding the <eqn-span>Z(<i>n</i>)</eqn-span> function and his numbering function, some people try this notion: <i>“Just leave out the informal parts of Gödel’s paper”</i> and they claim: <i>“the proof still follows from the formal parts of the paper”</i>. In fact, it doesn’t, since Gödel never furnished any formal proof whatsoever of his <a href="/ffgit/godel-original-english#PROP-V">Proposition&nbsp;5</a>, simply giving an informal outline of how he thought a proof might proceed. So if we leave out the informal parts of Gödel’s paper, then there isn’t any proof whatsoever - correct or incorrect - of his incompleteness result.</p>
<p>&nbsp;</p>

<h5>Weird idea 2. Gödel meant that there is another function involved</h5>
<p>This claim is often associated with a misunderstanding of the term “<span class="font80">NUMERAL</span>” where it occurs in the English translation by Jean Van Heijenoort, where in Relation 17, Van Heijenoort translates as:</p>
<p class="indent"><eqn-span>Z(<i>n</i>)</eqn-span> is the <span class="font80">NUMERAL</span> denoting the number <eqn-i>n</eqn-i>.</p>
<p class="line5">&nbsp;</p>
<p>Here it is important to understand that here the term “<span class="font80">NUMERAL</span>” is <i><b>not</b></i> used according to its meaning in English - Van Heijenoort uses the term “<span class="font80">NUMERAL</span>” in small capitals to indicate the Gödel number of “numeral”, which was previously defined to be the term that applies to any string of symbols of the formal system of the form <eqn-span><i>fff</i>&hellip;0</eqn-span>, so that what the above line is saying is that <eqn-span>Z(<i>n</i>)</eqn-span> is the Gödel number of the formal string <eqn-span><i>fff</i>&hellip;0</eqn-span> that is&nbsp;<eqn-i>n</eqn-i>. Van Heijenoort’s choice of the terms “numeral” and <span class="font80">NUMERAL</span> is rather unfortunate since the term “numeral” already has its own specific meaning in English and this has led many people to believe that there is some sort of association of the term with the usual English usage - a belief that is completely mistaken (this is why I chose the term “number-string” instead in my translation).</p>
<p>&nbsp;</p>
<p>The claim is that while the variable of the&nbsp;<eqn-span>Z</eqn-span> function has the domain of natural numbers, the restricted domain of the Gödel numbering function is formal sequences of the form <eqn-span><i>fff</i>&hellip;0</eqn-span> and which, according to the claim, are not numbers, so that Gödel actually meant:</p>
<p class="indent"><eqn-span>Z(<i>n</i>) = Φ(<i>s</i>)</eqn-span></p>
<p>and that Gödel meant that there is some sort of connection between the <eqn-i>n</eqn-i> in the <eqn-span>Z</eqn-span> function and the <eqn-i>s</eqn-i> in the Gödel numbering function. This, of course, is invoking an additional function that does not appear anywhere within Gödel’s paper, either explicitly or implicitly, so that the claim entails either:</p>
<p class="indent">(A) <eqn-span>Z(<i>n</i>) = Φ(<i>s</i>)</eqn-span>, where <eqn-span><i>n</i> = <i>function</i>&#8202;(<i>s</i>)</eqn-span>, i.e: <eqn-span>Z(&#8202;<i>function</i>&#8202;(<i>s</i>)) = Φ(<i>s</i>)</eqn-span></p>
<p>or</p>
<p class="indent">(B) <eqn-span>Z(<i>n</i>) = Φ(<i>s</i>)</eqn-span>, where <eqn-span><i>s</i> = <i>function</i>&#8202;(<i>n</i>)</eqn-span>, i.e: <eqn-span>Z(<i>n</i>) = Φ(&#8202;<i>function</i>&#8202;(<i>n</i>))</eqn-span></p>
<p class="line5">&nbsp;</p>
<p>where these imagined functions that are conjured out of nothing are often given a name that includes the word <i>“numeral”</i>.</p>
<p class="line12">&nbsp;</p>
<p>The obvious error in the claim is the fact that the formal sequences of the form <eqn-span><i>fff</i>&hellip;0</eqn-span> are, in fact, natural numbers since they satisfy the Peano axioms.</p>
<p class="line9">&nbsp;</p>
<p>Besides that, if the domain of <eqn-i>n</eqn-i> is natural numbers and the domain of <eqn-i>s</eqn-i> is not natural numbers, then (A) fails, since there the <eqn-span><i>function</i>&#8202;(<i>s</i>)</eqn-span> cannot be a number-theoretic function since, by the claim, its input values are supposedly not natural numbers, but sequences of the form <eqn-span><i>fff</i>&hellip;0</eqn-span>, and so it cannot be substituted for the free variable of the <eqn-span>Z</eqn-span> function. Similarly (B) fails since there the <eqn-span><i>function</i>&#8202;(<i>n</i>)</eqn-span> cannot be an expression of the formal system, since its input values are supposedly natural numbers, and by the claim, natural numbers supposedly do not belong to the formal system, and so it cannot be substituted for the free variable of the Gödel numbering function.<span class="footnote">
	You can see an example of this sort of muddled thinking in Mendelson’s <i>Introduction to Mathematical Logic</i> (Chapman & Hall/CRC, 6th edition, 2015). After his Proposition&nbsp;3.4, he describes a function <eqn-span class="overbar3"><i>n</i></eqn-span> 
that converts “natural numbers” into numbers in the format of the formal system. Then in his Proposition&nbsp;3.27, he defines a purely number-theoretic function <eqn-span>Num(&#8239;<i>y</i>&#8239;)</eqn-span> and asserts that this <eqn-span>Num(&#8239;<i>y</i>&#8239;)</eqn-span> is “the Gödel number of the expression <eqn-i><span class="overbar3">y</span></eqn-i>&#8202;”, i.e, that <eqn-span>Num(&#8239;<i>y</i>&#8239;) = Φ(&#8239;<span class="overbar3"><i>y</i></span>&#8239;)</eqn-span>. But <eqn-span class="overbar3"><i>y</i></eqn-span> is not an expression of the formal system. It is a function that cannot be expressed in the formal system; the fact that when its free variable is substituted by a valid value the result is a particular expression is irrelevant - that would be confusing a function and an instance of its results. For some strange reason logicians seem to have difficulty discerning an obvious conflation of levels of language.</span></p>
<p class="line9">&nbsp;</p>
<p>This conflation of language systems can be easily demonstrated, taking the notion (B) above that:</p>

<div class="xScroll">
<p class="indent"><eqn-span>Z(<i>n</i>) = Φ(&#8239;<i>function </i>(<i>n</i>))</eqn-span></p>
<p>Then we have that, for example:</p>
<p class="indent"><eqn-span><i>function</i>(4) = <i>fffff</i>&#8202;0</eqn-span>, but we also have that <eqn-span><i>function</i>(4) = <i>function</i>(3 + 1)</eqn-span></p>
<p>But then it is also the case that:</p>
<p class="indent"><eqn-span><i>fff</i>&#8202;0 *+* <i>f&#8202;</i>0 = <i>function</i>(3) *+* <i>function</i>(1) = <i>function</i>(3 + 1)</eqn-span></p>
</div>
<p>where <eqn-span>*+*</eqn-span> represents addition in the formal system.&nbsp;</p>
<p>&nbsp;</p>
<p>And so, as before, we then have that:</p>

<div class="xScroll">
<p class="indent"><eqn-span>Z(4) = Φ[ &#8239;<i>function</i> (4)] = Φ[ &#8239;<i>function</i>(3 + 1)],</eqn-span></p>
<p>but although:</p>
<p class="indent"><eqn-span><i>function</i>(3 + 1) = <i>function</i>(3) *+* <i>function</i>(1) = <i>fff</i>&#8202;0 *+* <i>f&#8202;</i>0</eqn-span></p>
<p>we have:</p>
<p class="indent"><eqn-span>Z(4) ≠ Φ[ &#8239;<i>function</i>(3) *+* <i>function</i>(1)]</eqn-span></p>
<p>and:</p>
<p class="indent"><eqn-span>Z(4) ≠ Φ[&#8239;<i>fff</i>&#8202;0 *+* <i>f&#8202;</i>0]</eqn-span>.</p>
</div>
<p>Again, we have a contradiction.</p>

<p>&nbsp;</p>
<p>The absurdity of claims of the above sort is mind-bogglingly obvious, yet the authors of such claims seem to be totally unaware of the nonsensical nature of the claims. The irony is that the idea that the variables of the <eqn-span>Z</eqn-span> function and the Gödel numbering function are inherently different actually points to the underlying cause of the contradiction - that the <eqn-span>Z</eqn-span> function and the Gödel numbering function belong to different levels of language, where the Gödel numbering function is, in fact in a meta-language to the <eqn-span>Z</eqn-span> function. This is analyzed in detail in my paper on Gödel’s proof, see <a href="/ffgit/godel-flaw-formal-paper#MetaAndNTR">Meta-Language and ‘Number-Theoretic Relations’</a>.</p>

<p>&nbsp;</p>
<h5>Weird idea 3. Gödel’s Proposition 5 does not involve Gödel numbering</h5>
<p>This involves the claim that Gödel numbering is not required for Gödel’s Proposition 5, that <i>“Gödel numbering does not intrinsically show up at all”</i> in that proposition. Apart from the patently obvious fact that Gödel did not provide a proof of his Proposition 5, instead assuming it to be obvious, the reality is that the statement of the proposition does require Gödel numbering, being:</p>
<p class="indent spacing2"><b>Proposition&nbsp;V</b>: To every recursive relation <eqn-span>R(<i>x</i><sub>1</sub> … <i>x</i><sub>n</sub>)</eqn-span> there corresponds an <eqn-i>n</eqn-i>-place <i>relation-string</i> <eqn-i>r</eqn-i>&nbsp;&hellip;</p>
<p>and by Gödel’s definition of <i>relation-string</i> this is:</p>
<p class="indent spacing2"><b>Proposition&nbsp;V</b>:To every recursive relation <eqn-span>R(<i>x</i><sub>1</sub> … <i>x</i><sub>n</sub>)</eqn-span> there corresponds a Gödel number <eqn-i>r</eqn-i>, where <eqn-i>r</eqn-i> is the Gödel number of a formula with <eqn-i>n</eqn-i> free individual variables ...&nbsp;&hellip;</p>
<p>which unambiguously refers to Gödel numbering, confirming that Gödel numbering is an intrinsic part of the proposition.</p>
<p>&nbsp;</p>
<h5>Inconsistent systems</h5>
<p>Only those who are unaware of basic tenets of mathematics (or those who deliberately ignore them) think that, by carefully choosing the terms and expressions used in an argument in order to avoid any <i><b>explicit</b></i> appearance of a contradiction, in some way serves to demonstrate that the system and its assumptions cannot result in any <i><b>implicit</b></i> contradiction - that is, that the system is inherently consistent. That, of course is nonsense, and if it <i><b>can</b></i> be demonstrated that the assumptions of the system can result in a contradiction, that means that the system is inconsistent and hence any result given by that system is mathematical gibberish. Note that Gödel’s failure to give a detailed argument for his Proposition&nbsp;5 helps to hide the fact that his assumptions regarding that proposition are intrinsically contradictory.</p>
<p>&nbsp;</p>
<p>Note that you can see further wacky ideas and my responses in the <!--chvignore--><a href="#CmtBox">comments section</a><!--/chvignore--> below.</p>
</div>
<p>&nbsp;</p>
<p>At this point we have to acknowledge that we cannot give a brief summary of the details of how this Proposition&nbsp;V is a confusion of different levels of language by the use of this false equivalence. This is because Gödel did not actually give a fully detailed proof of this crucial part of his incompleteness paper - he simply assumed that it was correct, only giving a rough outline of how a proof of that crucial part might proceed, saying in his paper: <i>“We content ourselves here with indicating the proof of this proposition in outline, since it offers no difficulties of principle and is somewhat involved.”</i>&#8239;<span class="footnote">
	The convicted paedophile ex-professor Peter Smith, although he is a staunch advocate of Gödel’s proof, acknowledges in his paper, <a href="https://logicmatters.net/igt/pdfs/Exposition2.pdf" target="_blank" rel="noopener noreferrer"><span class="nobr"><span class="pdf">PDF&nbsp;</span>Expounding</span> the First Incompleteness Theorem</a>, that:<br>
	<span class="indent il-block"><i>“Gödel only sketches a proof by induction on the complexity of the definition of the (characteristic function of the) property or relation in terms of definitions by composition and recursion grounding out in the trivial initial functions. The crucial step is just asserted - ‘the processes of definition&nbsp;… (substitution and recursion) can both be formally reproduced in the system&nbsp;P’&nbsp;… it certainly isn’t spelt out.”</i></span><br>
	<span class="spacing1 il-block">See also <a href="/ffgit/godel-without-tears">Peter Smith’s Gödel Without (Too Many) Tears - Or Not?</a> and <a href="/pdfs/ff_smith.pdf"><span class="nobr"><span class="pdf">PDF&nbsp;</span>A</span> Fundamental Flaw in an Incompleteness Proof by Peter Smith</a>.</span></span>
This failure to give a fully detailed proof is even more surprising in view of <a href="/ffgit/godel-quotes#Precision">Gödel’s insistence elsewhere on clarity and precision</a>.</p>
<p>&nbsp;</p>
<p>And today almost every mathematician/<wbr>logician seems not only content to make the same intuitive assumption, but they also try to avoid any mention of this assumption when they talk about Gödel’s proof, for example see <a href="/ffgit/pierce-on-godels-incompleteness">David Pierce on Gödel’s incompleteness proof</a>. But in mathematics, one should never let intuition triumph over strict logic. Stranger still is that you might expect that several mathematicians/<wbr>logicians would have tried to do a fully detailed proof of this crucial part, and published their results - but where are such results? This is why I give in my paper<span class="footnote">
	See <a href="/ffgit/godel-flaw-formal-paper">The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem</a>.</span>
a fully detailed account following Gödel’s suggested outline of his Proposition&nbsp;V - and how the incorrect assumption of equivalence of the <eqn-span>Z(<i>n</i>)</eqn-span> and <eqn-span>Φ(<i>n</i>)</eqn-span> functions is an indispensable requirement of that proposition. I have now added a section to the paper that gives a brief summary of the underlying illogical assumption of equivalence of the <eqn-span>Z(<i>n</i>)</eqn-span> and <eqn-span>Φ(<i>n</i>)</eqn-span> functions that the proof relies on, so that a reader can see in a few pages that the proof is flawed.⁠<span class="footnote">
	It is also easy to show that the conflation of levels of language inherent in the assumption of equivalence of the <eqn-span>Z(<i>n</i>)</eqn-span> and <eqn-span>Φ(<i>n</i>)</eqn-span> functions leads to a blatant contradiction in Gödel’s Proposition&nbsp;V, see the page <a href="/ffgit/godel-proposition-v">Gödel’s Proposition&nbsp;V</a>.</span>&nbsp;<span class="footnote">
Also see <a href="/ffgit/godel-guide-8#Zfunction">Gödel’s flawed assumption of equivalence</a>.</span></p>
<p>&nbsp;</p>
<p>It might be noted here that Gödel’s incompleteness paper is not the only paper where he made a completely unproven assertion which almost everyone assumed to be correct because of Gödel’s esteemed reputation. In that other paper Gödel made a similar intuitive assertion, but it was eventually proved (some 50 years later) that Gödel’s intuitive assumption stated precisely the <i><b>opposite</b></i> of what was actually the case. For details of this see <a href="/ffgit/godel-error-2">Gödel’s Intuitive Error No.2</a>. Not many mathematicians are aware of this - Gödel’s reputation is still so honored today that most discussions of Gödel’s work skip over this inconvenient detail.</p>
<p>&nbsp;</p>
<h3>A Simplified explanation of Gödel’s proof and the flaw in the proof</h3>
<p>If you are not familiar with the details of Gödel’s proof, I suggest that you either read the <a href="/ffgit/GodelSimplified0">simplified explanation of Gödel’s proof</a> on this website, or read the novel <a href="/ffgit/book_shackles_of_conviction">The&nbsp;Shackles&nbsp;of&nbsp;Conviction</a>. Both include an explanation of the <a class="tt_theorem tt" href="/ffgit/godels_theorem#theorem">theorem</a> and the flaw in Gödel’s proof of it in a simplified way that makes it accessible to the general reader.</p>
<p>&nbsp;</p>
<h3>A step by step walk-through guide to Gödel’s Proof</h3>
<p>For anyone who wishes to get to grips with Gödel’s original paper, I have created a <a href="/ffgit/godel-guide-0">walk-through guide to Gödel’s original incompleteness proof</a>, and which is intended to be read alongside the paper. Having looked for such a guide and could find none, I thought it odd that there were no such guides, considering how much other material has been written about Gödel’s proof. So I decided to create this guide to assist anyone else who wishes to become familiar with the details of Gödel’s argument.</p>
<p>&nbsp;</p>
<h3>A Detailed explanation of the flaw in Gödel’s proof</h3>
<p>On the other hand, if you are very familiar with the details of Gödel’s theorem, you will probably also want to see the details of the flaw in Gödel’s proof as covered in the paper:</p>
<p>&nbsp;</p>
<p><a href="/ffgit/godel-flaw-formal-paper">The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem</a></p>
<p>&nbsp;</p>
<p>The paper gives a comprehensive demonstration of the flaw in Gödel’s proof of his incompleteness theorem. The link above is a link to the latest version. Please note that the central argument of the paper remains unchanged throughout the revisions. The revisions have been to make the principles easier to understand. After some requests, the latest version includes a brief summary of the illogical unfounded assumption that the entire proof relies on, so that the reader can have an idea of the substance of the flaw within a few pages. If you want to see the previous versions to check that I have not changed the substance of my demonstration of the flaw in Gödel’s proof, please see the <a href="/jrm-site-map">site-map</a>. See also <a href="/ffgit/godels-1934-lectures">Gödel’s 1934 Undecidability lectures</a>, which, although by then Gödel had three years to reflect on his methods of proof, rather ironically allow a much simpler demonstration of the inherent confusion of language that is involved.</p>
<p>&nbsp;</p>
<h3>The Flaw and the Substitution function in Gödel’s proof</h3>
<p>The webpage <a href="/ffgit/godel_sb">Gödel’s Substitution Function</a> describes how Gödel’s incorrect use of a ‘substitution’ function confuses the systems that he is dealing with.</p>
<p>&nbsp;</p>
<h3>Responses to the demonstration of the flaw in Gödel’s proof</h3>
<p>Bertrand Russell	remarked:</p>
<p class="indent">&ldquo;<i>If a man is offered a fact which goes against his instincts, he will scrutinize it closely, and unless the evidence is overwhelming, he will refuse to believe it. If, on the other hand, he is offered something which affords a reason for acting in accordance to his instincts, he will accept it even on the slightest evidence</i>.&rdquo;&#8239;<span class="footnote">
	Bertrand Russell, in Ch. VI: International relations in <i>Roads To Freedom, </i>G. Allen &amp; Unwin&nbsp;1918.</span>	</p>
<p>&nbsp;</p>
<p>It might be said that Russell did not go far enough - I would rewrite it as:</p>
<p class="indent"><i>If a person is offered a fact which goes against their instincts and beliefs, they will carefully scrutinize the evidence to seek a flaw in it - but if they fail to find any flaw in it, they will not necessarily concede - instead they may simply ignore that evidence, and continue to refuse to believe it. If, on the other hand, they are offered something which in accordance with their instincts and beliefs, they will be inclined to accept it without any detailed examination of any available evidence.</i></p>
<p>&nbsp;</p>
<p>And in the 1950s, Leon Festinger coined the term cognitive dissonance, and remarked:<span class="footnote">
	Leon Festinger: <i>“When Prophecy Fails: A Social &amp; Psychological Study of a Modern Group that Predicted the Destruction of the World”</i>, Harper-Torchbooks, 1956, ISBN: 0-06-131132-4.
	<span class="marginTop5 il-block indent">The psychologist Carol Tavris has expanded on Festinger&rsquo;s work and has written a book on the subject (co-written with Elliot Aronson): “<a href="https://en.wikipedia.org/wiki/Mistakes_Were_Made_(but_Not_by_Me)" target="_blank" rel="noopener noreferrer"><i>Mistakes were made (but not by me): Why we justify foolish beliefs, bad decisions, and hurtful acts</i></a>”, Harcourt, 2007.</span></span>
</p>
<p class="indent"><i>A man with a conviction is a hard man to change. Tell him you disagree and he turns away. Show him facts or figures and he questions your sources. Appeal to logic and he fails to see your point. But suppose he is presented with evidence, unequivocal and undeniable evidence, that his belief is wrong: what will happen? The individual will frequently emerge, not only unshaken, but even more convinced of the truth of his beliefs than ever before.</i></p>
<p>&nbsp;</p>
<p>And this is precisely what has happened with Gödel’s proof - I have provided a detailed analysis which the vast majority of people reject, even though they cannot find any error in it, while on the other hand, they are perfectly content to accept Gödel’s Proposition&nbsp;V, a crucial assertion in Gödel’s proof, for which Gödel never provided any detailed proof (and neither has anyone else done so).</p>
<p>&nbsp;</p>
<p>If you would like to see some of the responses that I have received regarding the demonstration of the flaw in Gödel’s proof, please see <a href="/ffgit/godel_flaw-response">Responses to the demonstration of the flaw</a>.</p>
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<h4>Recently added</h4>
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	<li class="spacing2"><a href="/infinite/hausdorff-measure-of-point-sets">The Hausdorff Paradox</a></li>
	<li class="spacing2"><a href="/set-theory/set-theory-5-non-natural-set-theory#ContradictionNonNat">Set Theory’s Inherent Contradiction</a></li>
	<li class="spacing2"><a href="/infinite/lebesgue-cranks">Lebesgue Measure Cranks</a></li>
		<li class="spacing2"><a href="/ffgit/uwe-petersen-on-godel-flaw">Uwe Petersen on Gödel’s incompleteness proof</a></li>
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<h4>Set theory</h4>
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	<p>The page <a href="/infinite/decreasing-intervals">Understanding sets of decreasing intervals</a> explains why certain definitions of sets of decreasing intervals are inherently contradictory unless limiting conditions are included, and the page <a href="/infinite/understand-infinity-and-limits">Understanding Limits and Infinity</a> explains how the correct application of limiting conditions can eliminate such contradictions. The paper <a href="/pdfs/svc-sets-and-measure.pdf"><span class="nobr"><span class="pdf">PDF&nbsp;</span>On</span> Smith-Volterra-Cantor sets and their measure</a> has additional material which gives a more formal version.</p>
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	<p>How to set up a system for easy insertion or changing of footnotes in a webpage, see <a href="/otherstuff/easy-footnotes-for-web-pages">Easy Footnotes for Web Pages</a>, now simplified and improved.</p>
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	<h4>New section added to paper on Gödel’s flawed paper</h4>
	<p>After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details the confusion of levels of language. The paper can be seen at <a href="/ffgit/godel-flaw-formal-paper">The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem</a>.</p>
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	<h4><a href="/infinite/cantor-grundlagen">Cantor’s Grundlagen and associated papers</a></h4>
	<p>To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers published between 1879 and 1884 under the overall title of <i>Über unendliche lineare Punktmannig-faltigkeiten,</i> which were published between 1879 and 1884. I now have English translations of <a href="/infinite/cantor-uber-unendliche-1">Part&nbsp;1</a>, <a href="/infinite/cantor-uber-unendliche-2">Part&nbsp;2</a>, <a href="/infinite/cantor-uber-unendliche-3">Part&nbsp;3</a> and the major part, <a href="/infinite/cantor-grundlagen">Part&nbsp;5 (Grundlagen)</a>. There is also a new English translation of <a href="/infinite/cantor-1878-ein-beitrag">Cantor’s “A Contribution to the Theory of Sets”</a>.</p>
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	<h4><a href="/ffgit/brief-history-meta-mathematics">A brief history of meta-<span class="nobr">mathematics</span></a></h4>
	<p>A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for strict rigor.
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	<h4><a href="/otherstuff/easy-footnotes-for-web-pages">Easy Footnotes (Updated)</a></h4>
	<p>How to set up a system for easy insertion or changing of footnotes in a webpage, see <a href="/otherstuff/easy-footnotes-for-web-pages">Easy Footnotes for Web Pages</a>, now simplified and improved.</p>
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	<h4>New section added to paper on Gödel’s flawed paper</h4>
	<p>After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details the confusion of levels of language. The paper can be seen at <a href="/ffgit/godel-flaw-formal-paper">The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem</a>.</p>
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	<h4><a href="/infinite/cantor-grundlagen">Cantor’s Grundlagen and associated papers</a></h4>
	<p>To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers published between 1879 and 1884 under the overall title of <i>Über unendliche lineare Punktmannig-faltigkeiten,</i> which were published between 1879 and 1884. I now have English translations of <a href="/infinite/cantor-uber-unendliche-1">Part&nbsp;1</a>, <a href="/infinite/cantor-uber-unendliche-2">Part&nbsp;2</a>, <a href="/infinite/cantor-uber-unendliche-3">Part&nbsp;3</a> and the major part, <a href="/infinite/cantor-grundlagen">Part&nbsp;5 (Grundlagen)</a>. There is also a new English translation of <a href="/infinite/cantor-1878-ein-beitrag">Cantor’s “A Contribution to the Theory of Sets”</a>.</p>
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	<h4><a href="/ffgit/brief-history-meta-mathematics">A brief history of meta-<span class="nobr">mathematics</span></a></h4>
	<p>A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for strict rigor.
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		<button onclick="if (!window.__cfRLUnblockHandlers) return false; showSubNav('SubNav1'); return false;" data-cf-modified-774d55e95ec8df738f40bf88-=""><span>Logic and language: </span><span>How the failure to consider the limitations of language causes logical errors</span></button>	
		<button onclick="if (!window.__cfRLUnblockHandlers) return false; showSubNav('SubNav2'); return false;" data-cf-modified-774d55e95ec8df738f40bf88-=""><span>Gödel and Incompleteness: </span><span>How an error of intuition has triumphed over rational logic for almost 100 years</span></button>
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	<p class="nw_h2">Logic and Language</p>
	<a tabindex="0" href="/topics/logic-and-language"><p>Language, Logic and Intuition:</p><p>A brief summary of the philosophy underpinning this site</p></a>
	<a tabindex="0" href="/topics/logic-1"><p>Logic and Language:</p><p>Why logic is not independent of language</p></a>
	<a tabindex="0" href="/topics/formal_language"><p>Formal Language:</p><p>What a formal language is and what can be said within in and what can be said about it</p></a>
	<a tabindex="0" href="/ffgit/brief-history-meta-mathematics"><p>A brief history of meta-mathematics:</p>
	<p>How the field of meta-mathematics has made untenable and illogical assumptions</p></a>
	<a tabindex="0" href="/topics/natural-language-and-reality"><p>Natural Language and Reality:</p><p>How philosophers have managed to become befuddled by natural language</p></a>
	<a tabindex="0" href="/topics/russell-on-denoting"><p>On Denoting:</p><p>Bertrand Russell’s classic paper on dealing with entities that are named but do not exist</p></a>
	<a tabindex="0" href="/topics/thinking-and-being"><p>Thinking and Being:</p><p>How the philosopher Irad Kimhi conjures up imaginary difficulties regarding statements of non-existence</p></a>
	<a tabindex="0" href="/topics/mathproof"><p>Mathematical Proofs:</p><p>Despite claims to the contrary, the reality is that much of mathematics is not supported by logical proof</p></a>
	<a tabindex="0" href="/topics/pseudomath"><p>Pseudo-mathematics:</p><p>A derogatory term that mathematicians sometimes apply to the work of others while ignoring the logical inconsistencies in their own work</p></a>
	<a tabindex="0" href="/topics/computerproofs"><p>Computer Proofs:</p><p>Are computer proofs as infallible as their programmers claim?</p></a>
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	<p class="nw_h2">Gödel and Incompleteness</p>
	<a tabindex="0" href="/ffgit/godels_theorem"><p>Gödel’s proof of incompleteness:</p><p>A quick summary of the proof and related matters</p></a>
	<a tabindex="0" href="/ffgit/godel-original-english"><p>Gödel’s paper:</p><p>An online English translation of Gödel’s incompleteness proof with clickable cross-references</p></a>
	<a tabindex="0" href="/ffgit/GodelSimplified0"><p>Gödel’s proof simplified:</p><p>A simplified version of Gödel’s incompleteness proof that demonstrates the method used by Gödel</p></a>
	<a tabindex="0" href="/ffgit/godel-guide-0"><p>A guide to Gödel’s proof:</p><p>A step by step guide to Gödel’s proof for those wishing to investigate it in depth</p></a>
	<a tabindex="0" href="/ffgit/godel-contradiction"><p>Gödel’s contradiction:</p><p>The contradiction that is inherent in Gödel’s proof</p></a>
	<a tabindex="0" href="/ffgit/godel_flaw"><p>Gödel’s flaw:</p><p>The key flaw in Gödel’s incompleteness proof</p></a>
	<a tabindex="0" href="/ffgit/godel-and-ai"><p>AI systems agree:</p><p>There is a fundamental error in Gödel’s Incompleteness Proof</p></a>	
	<a tabindex="0" href="/ffgit/godel-flaw-formal-paper"><p>Formal Paper: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem:</p><p>This paper gives a detailed formal analysis of the flaw in Gödel’s paper</p></a>
	<a tabindex="0" href="/ffgit/godel-proposition-v"><p>Gödel’s Proposition&nbsp;V:</p><p>The crucial part of Gödel’s proof that he never actually proved, instead simply assuming it to be correct</p></a>
	<a tabindex="0" href="/pdfs/ff_representability.pdf"><p><span class="pdf">PDF&nbsp;</span>Representability:</p><p>Why claims that a formal system can logically refer to itself are erroneous</p></a>
	<a tabindex="0" href="/ffgit/godels-1934-lectures"><p>Gödel’s 1934 Undecidability lectures:</p><p>The confusion of language inherent in Gödel’s 1934 lectures</p></a>
	<p class="nw_h3">Other flawed incompleteness proofs</p>
	<a tabindex="0" href="/ffgit/kleene-rogers-incompleteness-errors"><p>Errors in incompleteness proofs by Kleene and Rogers:</p><p>How Stephen Kleene and Hartley Rogers make the same erroneous assumption to generate an incompleteness proof</p></a>
	<a tabindex="0" href="/ffgit/geb"><p>Gödel, Escher, Bach:</p><p>The flawed proof of incompleteness as given in the book <i>Gödel, Escher, Bach</i></p></a>
	<a tabindex="0" href="/ffgit/godel-without-tears"><p>Peter Smith’s Gödel Without (Too Many) Tears:</p><p>The error in Peter Smith’s incompleteness proof in his <i>Gödel Without (Too Many) Tears</i></p></a>
	<a tabindex="0" href="/pdfs/ff_smith.pdf"><p><span class="pdf">PDF&nbsp;</span>Peter Smith’s <i>An Introduction To Gödel’s Theorems</i>:</p><p>The error in Peter Smith’s incompleteness proof in his book <i>An Introduction To Gödel’s Theorems</i></p></a>
	<a tabindex="0" href="/ffgit/nagel-newman"><p>Nagel &amp; Newman</p><p>The error in Nagel &amp; Newman’s incompleteness proof</p></a>
	<a tabindex="0" href="/ffgit/analysis_inc_proofs"><p>Other incompleteness proofs:</p><p>Detailed analyses of several other incompleteness proofs</p></a>
	<a tabindex="0" href="/ffgit/errors_inc_proofs"><p>Common errors in incompleteness proofs:</p><p>Errors that can be shown to be common in many attempts at incompleteness proofs</p></a>
	<a tabindex="0" href="/ffgit/diagonal_lemma"><p>The “Diagonal Lemma”:</p><p>The key step in several incompleteness proofs and which contains an obvious error</p></a>
	<a tabindex="0" href="/ffgit/hilberts-tenth-problem"><p>Hilbert’s tenth problem</p><p>How a proof related to the problem is claimed to lead to a proof of incompleteness</p></a>
	<a tabindex="0" href="/blogs/blog-good-math-bad-math"><p>Good Math, Bad Math on Gödel:</p><p>An attempt at an explanation of Gödel’s incompleteness proof that falls down at the final hurdle</p></a>
	<a tabindex="0" href="/ffgit/aaronson1"><p>The Halting Problem and incompleteness proofs:</p><p>The naively simplistic assumptions used in attempts to prove incompleteness using Turing’s halting problem</p></a>
	<a tabindex="0" href="/ffgit/yet-another-flawed-proof"><p>Oh No&#8202;! Yet Another Flawed Incompleteness Proof:</p><p>A look at some incompleteness proofs that have very obvious errors</p></a>
	<a tabindex="0" href="/ffgit/incomplete-formal"><p>Formal papers:</p><p>Papers detailing errors in published claims of incompleteness proofs, including computer proofs</p></a>
	<p class="nw_h3">Other related matters</p>
	<a tabindex="0" href="/ffgit/uwe-petersen-on-godel-flaw"><p>Uwe Petersen on Gödel</p><p>An analysis of a critique of my work on Gödel’s incompleteness proof</p></a>
	<a tabindex="0" href="/ffgit/pierce-on-godels-incompleteness"><p>David Pierce and Gödel:</p><p>How mathematicians avoid mentioning the unproven assumption that Gödel used to obtain his incompleteness result</p></a>
	<a tabindex="0" href="/ffgit/godel-error-2"><p>Gödel’s other intuitive error:</p><p>An intuitive error by Gödel in another of his published papers, an error that was only discovered years later</p></a>
	<a tabindex="0" href="/ffgit/brief-history-meta-mathematics"><p>A brief history of meta-mathematics:</p><p>How the field of meta-mathematics has made untenable and illogical assumptions</p></a>
	<a tabindex="0" href="/ffgit/godel-quotes"><p>Statements by Gödel:</p><p>Some surprising statements by Gödel that reflect his Platonist philosophy</p></a>
	<a tabindex="0" href="/ffgit/godel_and_machines"><p>Gödel, Minds, and Machines:</p><p>A selection of claims for and against the notion that Gödel’s proof show that humans can deduce truths that machines cannot</p></a>
	<a tabindex="0" href="/blogs/blog-man-versus-machine"><p>Man versus Machine:</p><p>Roger Penrose’s bizarre claim that Gödel’s incompleteness proof implies that our brains must be using quantum processes that a machine cannot replicate</p></a>
	<a tabindex="0" href="/ffgit/godel_links"><p>Links regarding Gödel:</p><p>Links to various articles about Gödel’s proof</p></a>
	<a tabindex="0" href="/ffgit/godel_sb"><p>Gödel’s substitution function:</p><p>How Gödel uses substitution in his proof</p></a>
	<a tabindex="0" href="/ffgit/true_unprovable"><p>True but unprovable?</p><p>An examination of the claim that Gödel’s proof shows a formula to be true but unprovable in the given system</p></a>
	<a tabindex="0" href="/ffgit/intuitionism"><p>Intuitionism:</p><p>Intuition, Intuitionism and Gödel’s proof</p></a>
	<a tabindex="0" href="/ffgit/representability"><p>Representability:</p><p>A key claim in many attempts at incompleteness proofs is the claim that a certain expression has a precise representation in the formal system</p></a>
	<a tabindex="0" href="/ffgit/church1"><p>Church’s Theorem (An Unsolvable Problem):</p><p>How Church makes the same error as in many incompleteness proofs</p></a>
	<a tabindex="0" href="/ffgit/girard-blind-spot-ch2"><p>Jean-Yves Girard on Incompleteness:</p><p>A critical look at Jean-Yves Girard’s claims regarding Incompleteness from his book <i>The Blind Spot</i></p></a>
	<a tabindex="0" href="/ffgit/review-oconnor"><p>A critique of my paper:</p><p>A Review by Russell O’Connor of my paper on the flaw in Gödel’s proof</p></a>
	<a tabindex="0" href="/ffgit/show_error"><p>Flawed disproofs of Gödel’s proof:</p><p>Because Gödel’s proof appears paradoxical to many people, many people have attempted unsuccessfully to find the flaw in it</p></a>
	<a tabindex="0" href="/ffgit/book_shackles_of_conviction"><p>The Shackles of Conviction:</p><p>A novel based around finding the flaw in Gödel’s proof</p></a>
	<a tabindex="0" href="/ffgit/mathfiction1"><p>A review of my book:</p><p>A review of the book “The Shackles of Conviction” by Dr Kasman of MathFiction</p></a>
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	<p class="nw_h2">Infinity</p>
	<a tabindex="0" href="/infinite/infinity-1"><p>Infinity and the Infinite:</p><p>How conventional mathematics makes illogical assumptions that lead to contradictions</p></a>
	<a tabindex="0" href="/infinite/diagonal-proof"><p>Cantor’s 1891 Diagonal proof:</p><p>A complete logical analysis that demonstrates how several untenable assumptions have been made concerning the proof</p></a>
	<a tabindex="0" href="/infinite/enumerations-in-different-languages"><p>Non-Diagonal Proofs and Enumerations:</p><p>Why an enumeration can be possible <i>outside</i> of a mathematical system even though it is not possible <i>within</i> the system</p></a>
	<a tabindex="0" href="/infinite/cardinal-numbers"><p>Are there different sizes of infinity?</p><p>A look at the conventional definition of Cardinal Numbers and Cardinality</p></a>
	<a tabindex="0" href="/infinite/indefinable-numbers"><p>Indefinable numbers:</p><p>Why the notion of “indefinable numbers” is inherently contradictory</p></a>
	<a tabindex="0" href="/infinite/origins-of-transfinite"><p>The origins of transfinite numbers:</p><p>How Cantor started the notion of numbers bigger than a limitlessly large number</p></a>	
	<a tabindex="0" href="/infinite/goodstein"><p>Proving Goodstein’s theorem:</p><p>Are transfinite numbers really necessary to prove that every Goodstein sequence terminates?</p></a>
	<a tabindex="0" href="/infinite/banach-tarski-hausdorff-paradox"><p>The Banach-Tarski Paradox:</p><p>The real reason that gives rise to the paradox</p></a>		
	<a tabindex="0" href="/infinite/axiom-of-choice-well-ordering"><p>The Axiom of Choice and Well-Ordering:</p><p>A clarification of the ideas behind the terms “axiom of choice” and “well-ordering”</p></a>
	<a tabindex="0" href="/topics/platonism"><p>Platonism:</p><p>How Platonism has been the source of irrational assumptions about the infinite</p></a>
	<a tabindex="0" href="/infinite/continuum-hypothesis"><p>The Continuum Hypothesis:</p><p>How the notion of different levels of infinity created an unsolvable problem</p></a>
	<a tabindex="0" href="/infinite/proof-more-reals-than-naturals"><p>Proof of more real numbers than natural numbers?</p><p>Is there actually any proof that there are more real numbers than natural numbers?</p></a>
	<a tabindex="0" href="/infinite/courant-robbins"><p>The Courant &amp; Robbins Contradiction:</p><p>A contradiction in a proof by Courant and Robbins that demonstrates the inherent contradictions that result from illogical assumptions about the infinite</p></a>
	<a tabindex="0" href="/infinite/lebesgue-measure"><p>Lebesgue Measure Theory:</p><p>A logical analysis that demonstrates inherent contradictions in the Lebesgue theory of measure</p></a>
	<a tabindex="0" href="/infinite/lebesgue-cranks"><p>Lebesgue Measure Cranks:</p><p>A look at cranky attempts to justify Lebesgue measure theory</p></a>
	<a tabindex="0" href="/infinite/understand-infinity-and-limits"><p>Understanding Limits and Infinity:</p><p>How the correct application of limit states to sets composed of infinitely many intervals can eliminate contradictions.</p></a>
	<a tabindex="0" href="/infinite/decreasing-intervals"><p>Understanding sets of decreasing intervals:</p><p>Key points for understanding sets of decreasing intervals</p></a>
	<a tabindex="0" href="/infinite/real-numbers-and-language"><p>Real numbers and infinity:</p><p>An analysis of what real numbers are in terms of a logical consideration of language</p></a>
	<a tabindex="0" href="/infinite/cantor-invented-numbers"><p>Cantor’s invented numbers:</p><p>How Cantor’s invention of transfinite numbers ignored obvious contradictions</p></a>
	<a tabindex="0" href="/infinite/cantor-religion-transfinite"><p>Cantor’s religious beliefs:</p><p>How Cantor’s religious beliefs influenced his invention of transfinite numbers</p></a>
	<a tabindex="0" href="/infinite/list-with-no-diagonal"><p>A list of real numbers with no diagonal number:</p><p>How to define a list of real numbers for which there is no Diagonal number</p></a>
	<a tabindex="0" href="/infinite/cantors-1874-proof"><p>Cantor’s 1874 Proof:</p><p>A proof of non-denumerability preceding his better-known 1891 Diagonal Proof</p></a>
	<a tabindex="0" href="/infinite/actual-and-potential-infinity"><p>Actual and Potential Infinity:</p><p>Are there two types of infinity, actual completed infinity and potential infinity?</p></a>
	<a tabindex="0" href="/infinite/the-power-set-proof"><p>The Power Set Proof:</p><p>A proof based on the idea behind Cantor’s 1891 Diagonal Proof</p></a>
	<a tabindex="0" href="/topics/alexander-horned-sphere"><p>Alexander’s Horned Sphere:</p><p>A definition of a sphere with infinitely dividing horns: what it actually defines depends on the precise definition</p></a>
	<a tabindex="0" href="/topics/alexander-sphere-pierce"><p>Alexander’s Sphere according to Pierce:</p><p>A look at an article by David Pierce on the Alexander horned sphere</p></a>
	<a tabindex="0" href="/infinite/why-do-people-believe-weird-things"><p>Why do people believe weird things?</p><p>Why do intelligent people continue to defend beliefs that have no logical basis?</p></a>
	<a tabindex="0" href="/infinite/infinite-sums-1"><p>Sums of infinitely many fractions, part one:</p><p>How there is an inherent contradiction inherent in the concept that infinitely many numbers can be summated</p></a>
	<a tabindex="0" href="/infinite/infinite-sums-2"><p>Sums of infinitely many fractions, part two:</p><p>Another demonstration of an inherent contradiction in the concept that infinitely many numbers can be summated</p></a>
	
<a tabindex="0" href="/infinite/book-the-infinity-delusion"><p>The Infinity Delusion:</p><p>A book looking at how infinity is treated by modern mathematics and pointing out the logical fallacies involved</p></a>
	<a tabindex="0" href="/infinite/lavine-misunderstanding"><p>Misunderstanding the Infinite:</p><p>A review of Shaughan Lavine’s book <i>Understanding the Infinite</i></p></a>
	<a tabindex="0" href="/infinite/hilbert-uber-das-unendliche"><p>On the Infinite:</p><p>A critical look at David Hilbert’s 1925 lecture <i>On the Infinite</i> (Über das Unendliche)</p></a>
	<a tabindex="0" href="/infinite/infinite-formal"><p>Formal papers:</p><p>A selection of formal papers related to the infinite</p></a>

	<p class="nw_h3">Online English translations</p>
	<a tabindex="0" href="/infinite/cantor-uber-unendliche-1"><p>Translation: On infinite linear point-sets, Part&nbsp;1</p><p><span class="indent2 block">(Über unendliche lineare Punktmannig-faltigkeiten)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-uber-unendliche-2"><p>Translation: On infinite linear point-sets, Part&nbsp;2</p><p><span class="indent2 block">(Über unendliche lineare Punktmannig-faltigkeiten)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-uber-unendliche-3"><p>Translation: On infinite linear point-sets, Part&nbsp;3</p><p><span class="indent2 block">(Über unendliche lineare Punktmannig-faltigkeiten)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-grundlagen"><p>Translation: Foundations of a general theory of sets</p><p><span class="indent2 block">(Grundlagen einer allgemeinen Mannigfaltigkeitslehre: This is Part&nbsp;5 of Über unendliche lineare Punktmannig-faltigkeiten)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-1874-uncountability-proof"><p>Translation: Cantor’s 1874 paper “On a Property of the Set of all Real Algebraic Numbers”</p><p><span class="indent2 block">(Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-1878-ein-beitrag"><p>Translation: Cantor’s 1878 paper “A Contribution to the Theory of Sets”</p><p><span class="indent2 block">(Ein Beitrag zur Mannigfaltigkeitslehre)</span></p></a>
	<a tabindex="0" href="/infinite/cantors-original-1891-proof"><p>Translation: Cantor’s 1891 Diagonal paper “On an elementary question of set theory”</p><p><span class="indent2 block">(Über eine elemtare Frage de Mannigfaltigkeitslehre)</span></p></a>
	<a tabindex="0" href="/infinite/cantor-contributions-transfinite"><p>Translation: Cantor’s 1895 “Contributions to the Foundations of the Theory Of Transfinite Numbers”</p><p><span class="indent2 block">(Beiträge zur Begründung der transfiniten Mengenlehre)</span></p></a>
	<a tabindex="0" href="/infinite/hausdorff-measure-of-point-sets"><p>Felix Hausdorff&#8202;'s <i>A Remark on the Measure of Point Sets</i>&#8239;:</p><p>An online English translation of Felix Hausdorff&#8202;'s 1914 paper that led to the Banach-Tarski Paradox</p></a>
	<a tabindex="0" href="/infinite/vitali-paper-measure-points-on-line"><p>Giuseppe Vitali’s <i>On the problem of measuring sets of points</i>&#8239;:</p><p>An online English translation of Giuseppe Vitali’s 1905 paper that raised the problem of immeasurable sets of points</p></a>
	<a tabindex="0" href="/infinite/konig-on-foundations-english"><p>Julius König’s <i>On the Foundations of Set Theory</i>&#8239;:</p><p>An online English translation of the papers (Parts 1 and 2) that initiated the notion that there must be undefinable real numbers</p></a>
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	<p class="nw_h2">Set Theory</p>
	<a tabindex="0" href="/set-theory/set-theory-1-overview-set-theory">
	<p>Different types of set theories:</p><p>How mathematics forgot the lessons of the past when trying to develop a theory of sets</p></a>
	<a tabindex="0" href="/set-theory/set-theory-2-russells-paradox"><p>Russell’s paradox:</p><p>How Russell’s paradox showed that Cantor’s set theoretical ideas were fatally flawed</p></a>
	<a tabindex="0" href="/set-theory/set-theory-3-natural-set-theory"><p>Natural Set Theory:</p>
	<p>What a Natural Set Theory is and why it is very different to “naive” set theory</p></a>
	<a tabindex="0" href="/set-theory/set-theory-5-non-natural-set-theory"><p>Non-Natural Set Theory:</p><p>Why non-natural set theoretical ideas are the root of all problems in set theories</p></a>
	<a tabindex="0" href="/set-theory/set-theory-4-the-empty-set"><p>The ‘Empty Set’:</p><p>Why the notion of an ‘Empty Set’ generates more problems than it solves</p></a>
	<a tabindex="0" href="/set-theory/set-theory-6-myth-of-set-theory"><p>Foundations of Mathematics:</p><p>Is Set Theory really the Theory of Everything for Mathematics for all time?</p></a>	
	<a tabindex="0" href="/set-theory/set-theory-7-containers"><p>Containers:</p><p>The difference between containers and sets</p></a>
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	<p class="nw_h2">Platonism</p>
	<a tabindex="0" href="/topics/platonism"><p>Platonism:</p><p>How Platonism has been the source of irrational unfounded assumptions about the infinite</p></a>
	<a tabindex="0" href="/topics/myths-of-platonism"><p>The Myths of Platonism:</p><p>The assumption that certain supernatural things exist and the assertion that humans can accurately describe these things even though humans have no way of confirming such descriptions</p></a>
	<a tabindex="0" href="/topics/numbers-unicorns"><p>Numbers, chairs and unicorns:</p><p>The Platonist claim that numbers have an actual existence that is as real as the existence of chairs, rather than being human constructs like unicorns</p></a>
	<a tabindex="0" href="/topics/platonism-logical-blunder"><p>Platonism’s Logical Blunder:</p><p>The fallacy involved in the attempt for total abstraction</p></a>
	<a tabindex="0" href="/topics/indispensability-argument"><p>The Indispensability Argument:</p><p>A claim that tries to prove that mathematical entities are real things that exist independently of any human concepts</p></a>
	<a tabindex="0" href="/blogs/blog-moderate-platonism"><p>Moderate Platonism:</p><p>Gideon Rosen and John Burgess claim that their version of Platonism is ‘moderate’</p></a>
	<a tabindex="0" href="/topics/balaguer-platonism"><p>Mark Balaguer on Platonism:</p><p>Balaguer claims that there is no good argument against Platonism</p></a>
	<a tabindex="0" href="/blogs/blog-descartes-platonism"><p>Descartes’ Platonism:</p><p>A look at some Platonist aspects of Descartes’ philosophy</p></a>
	<a tabindex="0" href="/blogs/blog-perssons-platonism"><p>Ulf Persson on Platonism:</p><p>An impassioned apology in support of Platonism</p></a>
	<a tabindex="0" href="/paradoxes/platonist-rod-paradox"><p>The Platonist Rod Paradox:</p><p>How a Platonist mindset can take precedence over logic</p></a>
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	<p class="nw_h2">Paradoxes</p>
	<a tabindex="0" href="/paradoxes/paradoxes"><p>Paradoxes:</p><p>A quick overview of paradoxes and how they can arise by confusion of language</p></a>
	<a tabindex="0" href="/paradoxes/liar"><p>The Liar Paradox:</p><p>The original self-referential paradox: <i>“This sentence is false”</i></p></a>
	<a tabindex="0" href="/paradoxes/surprise-examination"><p>The Surprise Examination paradox:</p><p>A seemingly challenging paradox that is easily resolved by logical analysis.</p></a>
	<a tabindex="0" href="/paradoxes/balls-in-the-urn-paradox"><p>The Balls in the Urn Paradox (The Ross-Littlewood paradox):</p><p>How woolly thinking about infinity can lead to contradictions</p></a>
	<a tabindex="0" href="/paradoxes/platonist-rod-paradox"><p>The Platonist Rod Paradox:</p><p>Another paradox inspired by the Balls in the Urn Paradox</p></a>
	<a tabindex="0" href="/paradoxes/drinker"><p>The Drinker Paradox:</p><p>Based on the pretense that one can apply classical formal logic to a constantly changing real world situation</p></a>
	<a tabindex="0" href="/paradoxes/church-godel"><p>Gödel’s Church Paradox:</p><p>A paradox that is reliant on the confusion of different levels of language</p></a>
	<a tabindex="0" href="/paradoxes/curry-paradox"><p>Curry’s Paradox (Lob’s paradox):</p><p>A paradox that relies on an inconsistent system for its creation</p></a>
	<a tabindex="0" href="/paradoxes/goodman-paradox"><p>Goodman’s Paradox (The New Riddle of Induction):</p><p>Goodman claims that his paradox arises from scientific considerations, whereas it is a result of ambiguity of definition</p></a>
	<a tabindex="0" href="/paradoxes/bertrand-paradox"><p>The Bertrand Paradox:</p><p>How you can have differing but correct answers to a question that isn’t sufficiently well-defined</p></a>
	<a tabindex="0" href="/paradoxes/wang-paradox"><p>Wang’s Paradox (The Sorites paradox):</p><p>When you remove grains of sand from a heap of sand, when does the heap stop being a heap?</p></a>
	<a tabindex="0" href="/paradoxes/grim-reaper-benardete-paradox"><p>The Grim Reaper Paradox (Benardete’s Paradox):</p><p>Does the notion of an infinite sequence of killer reapers have any significance?</p></a>
	<p class="nw_h3">Other Paradoxes based on self-reference</p>
	<a tabindex="0" class="indent2" href="/paradoxes/richard-paradox"><p>Richard’s Paradox</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/smullyan-paradox"><p>Smullyan’s paradoxes</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/grelling-nelson"><p>Grelling-Nelson Paradox</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/quine-paradox"><p>Quine’s Paradox</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/yablo"><p>Yablo’s Paradox</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/berry-paradox"><p>Berry Paradox</p><p class="displayNone">&nbsp;</p></a>
	<a tabindex="0" class="indent2" href="/paradoxes/halbach-zhang-yablo-without-godel"><p>Halbach and Zhang’s <i>Yablo without Gödel</i></p><p class="displayNone">&nbsp;</p></a>
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	<p class="nw_h2">Archive</p>
	<a tabindex="0" href="/blogs/blog-failure-to-understand"><p>A failure to understand:</p><p>How to completely misunderstand Cantor’s Diagonal proof</p></a>
	<a tabindex="0" href="/blogs/blog-cranks-and-crackpots"><p>Cranks and Crackpots:</p><p>How experts fail to distinguish cranks and crackpots from serious researchers</p></a>
	<a tabindex="0" href="/blogs/blog-random-numbers"><p>Random selection:</p><p>What does it mean to <i>“select a real number at random”</i>?</p></a>
	<a tabindex="0" href="/blogs/blog-moderate-platonism"><p>Moderate Platonism:</p><p>Gideon Rosen and John Burgess claim that their version of Platonism is ‘moderate’</p></a>
	<a tabindex="0" href="/blogs/blog-descartes-platonism"><p>Descartes’ Platonism:</p><p>A look at the Platonist aspects of Descartes’ philosophy</p></a>	
	<a tabindex="0" href="/blogs/blog-consciousness-myth-dead-end"><p>The Dead End Path of the Consciousness Myth:</p><p>Why the vague and nebulous notion behind the term “consciousness” is a path that leads to nowhere</p></a>	
	<a tabindex="0" href="/blogs/blog-perssons-platonism"><p>Ulf Persson on Platonism:</p><p>A eulogy on Platonist beliefs, short on logic and replete with fallacious arguments</p></a>
	<a tabindex="0" href="/blogs/blog-is-math-effective"><p>Is Mathematics Unreasonably Effective?</p><p>Why the argument that conventional mathematics is amazingly effective for science is completely wrong</p></a>
	<a tabindex="0" href="/blogs/blog-duplicity-mark-chu-carroll"><p>The duplicity of Mark Chu-Carroll:</p><p>How a mathematician refuses to play fair in an argument</p></a>
	<a tabindex="0" href="/blogs/blog-searle-inanity"><p>A John Searle inanity:</p><p>A demonstration of the obvious error in John Searle’s argument that human consciousness might be something that is non-computable</p></a>
	<a tabindex="0" href="/blogs/blog-man-versus-machine"><p>Man versus Machine:</p><p>Roger Penrose’s bizarre claim that because of Gödel’s incompleteness proof our brains must be using quantum processes that a machine cannot replicate</p></a>
	<a tabindex="0" href="/blogs/blog-fake-mathematics"><p>Fake News and Fake Mathematics:</p><p>How some parts of mathematics have been fabricated without any supporting logical basis</p></a>
	<a tabindex="0" href="/blogs/blog-ned-blocks-blockhead"><p>Ned Block’s Blockhead:</p><p>An attempt to use the concept of a magical non-intelligent machine to ‘prove’ that a non-intelligent machine can pass any Turing type test</p></a>
	<a tabindex="0" href="/blogs/blog-are-we-alone"><p>Are we alone in the Universe?</p><p>Examining the claim that Gödel’s incompleteness proof is the most profound proof in the entire universe, and so we should broadcast it to alien intelligences</p></a>
	<a tabindex="0" href="/blogs/blog-bishop-dancing-pixies"><p>Bishops Dancing with Pixies?</p><p>J. Mark Bishop’s erroneous claim that Artificial Intelligence can never match nor surpass human intelligence</p></a>
	<a tabindex="0" href="/blogs/blog-2015-artificial-intelligence"><p>Artificial Intelligence:</p><p>A look at possible futures in view of the rise of artificial intelligence</p></a>
	<a tabindex="0" href="/blogs/blog-chinese-room"><p>The Chinese Room:</p><p>John Searle’s fallacious argument that a machine could not possibly understand a human language</p></a>
	<a tabindex="0" href="/blogs/blog-marys-room-fallacy"><p>The Mary’s Room fallacy:</p><p>A fallacious argument that attempts to prove that consciousness cannot be the result of purely physical processes</p></a>
	<a tabindex="0" href="/blogs/blog-mind-body-problem"><p>The Mind-Body problem:</p><p>Why there is no Mind-Body problem</p></a>
	<a tabindex="0" href="/blogs/blog-good-math-bad-math"><p>Good Math, Bad Math on Gödel:</p><p>An attempt at an explanation of Gödel’s incompleteness proof that falls down at the final hurdle</p></a>
	<a tabindex="0" href="/blogs/blog-crackpot-kalimuthu"><p>Kalimuthu Sennimalai:</p><p>Is he a crank or a comedian?</p></a>
	<a tabindex="0" href="/blogs/blog-belief-systems-versus-rationality"><p>Belief systems versus rationality:</p><p>Common factors between different belief systems</p></a>
	<a tabindex="0" href="/blogs/blog-chu-carroll-formal-maths"><p>Mark Chu-Carroll on Cantor:</p><p>A poor attempt to defend some of Cantor’s ideas</p></a>
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	<p class="nw_h2">Other Topics</p>
	<a tabindex="0" href="/ffgit/brief-history-meta-mathematics"><p>A brief history of meta-mathematics:</p><p>How the field of meta-mathematics has made untenable and illogical assumptions</p></a>
	<a tabindex="0" href="/topics/computerproofs"><p>Computer Proofs:</p><p>Must a proof be correct simply because someone claims that it has been checked by a computer program?</p></a>
	<a tabindex="0" href="/topics/crackpot-john-gabriel"><p>How to tell if someone is a crackpot:</p><p>An examination of the claims of the crackpot John Gabriel</p></a>
	<a tabindex="0" href="/blogs/blog-crackpot-kalimuthu"><p>Kalimuthu Sennimalai - Crackpot or Joker?:</p><p>An examination of a claimed “proof&#8202;” by Kalimuthu Sennimalai</p></a>
	<a tabindex="0" href="/infinite/why-do-people-believe-weird-things"><p>Why do people believe weird things?</p><p>Why do intelligent people continue to defend beliefs that have no logical basis?</p></a>
	<a tabindex="0" href="/topics/novakant"><p>A discussion:</p><p>A typical discussion about my work</p></a>
	<a tabindex="0" href="/topics/chaitins-omega"><p>Chaitin’s <i>Omega number</i>&#8239;:</p><p>The inherent error in Chaitin’s claims about his ‘Omega number’</p></a>
	<a tabindex="0" href="/topics/turings-uncomputable-number"><p>Turing’s <i>Uncomputable Number</i>&#8239;:	</p><p>An error in Turing’s claim that he has defined an ‘uncomputable number’</p></a>
	<a tabindex="0" href="/topics/dobbs-jackson-roe-wade"><p>Dobbs vs Jackson &amp; Roe vs Wade:</p><p>A look at the recent US supreme Court’s ruling re abortion</p></a>
	<a tabindex="0" href="/topics/the-consistency-of-arithmetic"><p>The Consistency of Arithmetic:</p><p>An erroneous proof of consistency by Storrs McCall</p></a>
	<a tabindex="0" href="/topics/excluded-middle"><p>The Law of the Excluded Middle:</p><p>Is it being applied within the system or outside of the system?</p></a>
	<a tabindex="0" href="/topics/quora-comments"><p>Responses to Quora questions:</p><p>My responses to some questions and answers posted on the censorial site Quora</p></a>
	<a tabindex="0" href="/topics/straight-and-crooked-thinking"><p>“Straight and Crooked Thinking”</p><p>An online copy of the book by Robert Thouless on fallacious and dishonest arguments.</p></a>
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