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SUMMARY:Saeed Salehi (University of Tabriz)
DTSTART:20221124T000000Z
DTEND:20221124T013000Z
DTSTAMP:20260414T085656Z
UID:Category_Theory/43
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Category_The
 ory/43/">Self-Reference and Diagonalization: their difference and a short 
 history.</a>\nby Saeed Salehi (University of Tabriz) as part of New York C
 ity Category Theory Seminar\n\n\nAbstract\nWhat is now called the Diagonal
  (or the Self-Reference) Lemma\,  is the statement that for every formula 
 <span style="font-family:Times New Roman"><i>F</i></span>(<i>x</i>)\, with
  the only free variable <i>x</i>\, there exists a sentence <i>&sigma\;</i>
  such that <i>&sigma\;</i> is equivalent to the <span style="font-family:T
 imes New Roman"><i>F</i></span> of the <font style="font-variant: small-ca
 ps">G&ouml\;del</font> code of <i>&sigma\;</i>\, i.e.\, <i>&sigma\;</i> <b
 >&equiv\;</b> <span style="font-family:Times New Roman"><i>F</i></span>(<b
 >#</b><i>&sigma\;</i>)\; and this equivalence is provable in certain weak 
 arithmetics. This lemma is credited to <font style="font-variant: small-ca
 ps">G&ouml\;del</font> (1931)\, in the special case when <i>F</i> is the <
 i>un</i>provability predicate\, and to <font style="font-variant: small-ca
 ps">Carnap</font> (1934) in the more general case.\n<br><br>\nIn this talk
 \, we will argue that <font style="font-variant: small-caps">G&ouml\;del</
 font>-<font style="font-variant: small-caps">Carnap</font>'s original Diag
 onal Lemma is not the modern formulation and was more similar to\, but not
  exactly identical with\, the Strong Diagonal (or Direct Self-Reference) L
 emma. This lemma\, so-called recently\, says that for every formula <span 
 style="font-family:Times New Roman"><i>F</i></span>(<i>x</i>)\, in a suffi
 ciently expressive language\, there exists a sentence <i>&sigma\;</i> such
  that <i>&sigma\;</i> is equal to the <span style="font-family:Times New R
 oman"><i>F</i></span> of the <font style="font-variant: small-caps">G&ouml
 \;del</font> code of <i>&sigma\;</i>\, i.e.\, <i>&sigma\;</i> <b>=</b> <sp
 an style="font-family:Times New Roman"><i>F</i></span>(<b>#</b><i>&sigma\;
 </i>)\; and this equality is provable in sufficiently strong theories. We 
 will attempt at tracking down the first appearance of the modern formulati
 on of the Diagonal Lemma in the equivalent form\, also in the strong direc
 t form of equality.\n
LOCATION:https://researchseminars.org/talk/Category_Theory/43/
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