Timeline for Looking for a source for Intended Interpretation
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2016 at 14:19 | comment | added | Mikhail Katz | ...interpretation in terms of the familiar integers people know about before they learn any formal mathematics, whether PA or ZFC. I am looking for a published source for such a comment. I am not sure what you have such great difficulty understanding here precisely. | |
| Mar 9, 2016 at 14:18 | comment | added | Mikhail Katz | @FrançoisG.Dorais, my comment about the constructions of the integers in a set-theoretic context by von Neumann and Zermelo was precisely meant to point out that the Intended Interpretation cannot be understood in a set-theoretic context because that would make it vacuous: the theory is interpreted in a set-theoretic context by assigning to the symbols 1,2,3 their set-theoretic meaning. This is vacuous because this is true in any interpretation and not specifically the Intended Interpretation, so obviously this is not what Wang meant. Rather, what he seems to have meant is an informal... | |
| Mar 9, 2016 at 13:22 | comment | added | François G. Dorais | @katz, my comment about ordinals is in direct response to your previous comment about von Neumann. It's about what you said, though I have since given up on understanding what you are asking. As for Wang, he said "originally intended" which can be understood as a reference to Peano and Dedekind, but can also be understood as what the author was intending earlier in the text. Now, I'll leave you to debate that with yourself. | |
| Mar 9, 2016 at 7:51 | comment | added | Mikhail Katz | ... Am I misinterpreting your comment about finite ordinals? | |
| Mar 9, 2016 at 7:51 | comment | added | Mikhail Katz | @FrançoisG.Dorais, I am not sure I understand your comment fully but at any rate Wang could not have meant that according to the "intended interpretation" these are finite ordinal numbers, because that would imply a set-theoretic interpretation where necessarily $1,2,3$ would be finite ordinal numbers, regardless of whether the interpretation in question is intended or not. On this reading Wang's comment would be vacuous. This question is based on the assumption that Wang is a distinguished logician and must have meant something by his comment that's not vacuous... | |
| Mar 8, 2016 at 7:46 | comment | added | Mikhail Katz | Dorais, to repeat, I am looking for a published source saying what the distinguished logician Wang said at the encyclopedia britannica entry. If you want to get me into the Intended Intepretation itself this would have to be at a different question. | |
| Mar 7, 2016 at 19:18 | comment | added | François G. Dorais | @katz, to repeat, if by "counting numbers" you mean "finite ordinals" then you should say so. | |
| Mar 7, 2016 at 18:08 | comment | added | Mikhail Katz | Whatever Wang refers to as the ordinary numbers. François, it is interesting that in your answer you seem to know what Peano and Dedekind meant by counting numbers--even though they apparently didn't have access to either von Neumann's or Zermelo's construction :-) | |
| Mar 7, 2016 at 18:01 | comment | added | François G. Dorais | @katz, if your "counting numbers" mean something other than "finite cardinals" then I must concur with others that your question is very unclear. | |
| Mar 7, 2016 at 17:46 | comment | added | Mikhail Katz | @SemenKutateladze, sounds promising. However, this seems unrelated to Fracois Dorais' answer. Would you care to format this as a separate answer? | |
| Mar 7, 2016 at 17:42 | comment | added | Semen Kutateladze | John Concoran used the term in Ancient Logic and Its Modern Interpretations: Proceedings of the Buffalo Symposium on Modernist Interpretations of Ancient Logic, 21 and 22 April, 1972 | |
| Mar 7, 2016 at 13:17 | comment | added | Mikhail Katz | ... but certainly today it is no novelty that Peano axioms can be deduced if one works in second order logic. At no point does Heck make a claim to the effect that this constitutes a proof of a claim that the Peano axioms allegedly describe the counting numbers. Indeed Heck does not mention such numbers in his article. I think your summary of Heck's claim is overstated. | |
| Mar 7, 2016 at 13:15 | comment | added | Mikhail Katz | Thanks for an interesting answer. I looked through R. Heck's article on Frege which also looks interesting. He examines the role of the problematic Axiom 5 (entailing an inconsistency) and of Hume's principle (which seems to amount to an enumeration of the scope of a predicate) and concludes that Frege's derivation of the Peano axioms does not depend on Axiom 5 but rather only on Hume's principle. As Heck himself says in his conclusion, this amounts to a reconstruction of Frege's argument in second order logic. Now it is interesting that Frege was able to do this so early on... | |
| Mar 6, 2016 at 20:17 | history | edited | François G. Dorais | CC BY-SA 3.0 |
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| Mar 6, 2016 at 20:02 | history | edited | François G. Dorais | CC BY-SA 3.0 |
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| Mar 6, 2016 at 19:56 | history | answered | François G. Dorais | CC BY-SA 3.0 |