Talk:True arithmetic
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foundations
[edit]The use of usual in the original definition shows that the set N and hence the structure N' are undefined (here). One can construct an N in Set Theory. However, some concepts about N are probably used in setting up Set Theory. There is a foundational problem involved that should at least be mentioned in the article. G. Blaine (talk) 17:51, 3 August 2017 (UTC)
Dependence on metatheory
[edit]It seems to me that we could add in the page that "true arithmetic" is not really unique: it depends on the metatheory one works in. Thus saying « true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers » is misleading. For instance if our metatheory were inconsistent we could prove that 0=1 in "true arithmetic" but if it is consistent (presumably ZF for instance) then we cannot prove this arithmetic sentence; thus true arithmetic depends on the metatheory. Another example is taking a Gödel encoding (call it of "ZF is consistent" in the language of arithmetic, this is possible as ZF is recursively axiomatized. Now and differ on some arithmetic statements (assuming ZF is consistent). I think one can formulate as a diophantine equation , with an explicit polynomial , for a more concrete realization of the above. This is an incontestably arithmetic statement which will be true or false depending on the metatheory one chooses; the above metatheories are somewhat contrived so as to produce such ambiguous arithmetic statements, but it seems that we could obtain similar ambiguous statements (what we could call metatheory-dependent arithmetic truths/statements) by comparing "true arithmetics" obtained from reasonable set theories actually used by set theorists, used as metatheories. I would have to think about it a little more to find more striking examples, worked out in careful details, but i think that from the perspective above there is no single "true arithmetic" but rather some conventional truths in arithmetic, depending on the mathematical community's practice, their preferred choice of axioms; and that the incompleteness of any set theory we currently use as metatheory leads to a certain type of "incompleteness" of "true arithmetic" -of course not in the usual logical sense of "incomplete", so we can call this new property "mathematical practice-incomplete" or "meta-incomplete". If i don't edit the page, despite my criticism, it is because i would prefer to understand better the phenomenon and ask the opinion of some logicians/set theorists, to see if they find this nuance congenial, useful. Plm203 (talk) 13:44, 27 August 2023 (UTC)