Timeline for What is the "reason" for modularity results?
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| May 22, 2018 at 0:38 | comment | added | Qfwfq | Uhm... very late to the party, but: @Charles Matthews, the expression "number theory" is clearly intended to mean different things in different contexts. When logicians talk about "number theory" they mostly mean the study of formal/syntactic theories of elementary arithmetic and their models; while when algebraists and number theorists say "number theory", they (or you!) really mean algebraic number theory + analytic number theory + maybe arithmetic geometry. They are really looking at different mathematical objects (probably with nonempty intersection in few aspects). | |
| Nov 18, 2011 at 19:39 | comment | added | Charles Matthews | Shrug. You can treat my comment as a straw man if you insist. It doesn't mention "provability" at all. If you want a paraphrase of the whole thought, it would be that the hierarchy of results that matters to the mainstream tradition of number theory is no kind of logical hierarchy. | |
| Nov 18, 2011 at 17:18 | comment | added | Emil Jeřábek | Well, neither Hofstadter nor Wolfram is a logician, actually. Anyway, you should read “a subset of” before any occurrence of “number theory” on that page (as is blatantly obvious from the Presburger arithmetic example: no one in their right mind would claim that all of number theory can be formulated in a system whose expressive power is limited to Boolean combinations of linear inequalities with integer coefficients and congruences). It is not an explanation of what number theory is in terms of a formal system, but vice versa. | |
| Nov 16, 2011 at 16:37 | comment | added | Charles Matthews | This link uses it in that kind of fashion: mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html Of course the link there is to number theory as "the higher arithmetic" instead. While I can't prove that a logician wrote that text, it's what I meant. | |
| Nov 15, 2011 at 11:38 | comment | added | Emil Jeřábek | Being a logician myself, I can’t recall anyone equating number theory with provability in Peano arithmetic (or any other formal theory of arithmetic for that matter). That sounds like a curious misconception. Could you be more specific? | |
| Sep 14, 2011 at 13:38 | history | answered | Charles Matthews | CC BY-SA 3.0 |