User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-27513631-20180611232757/@comment-27513631-20180613082942

"What "actual" means here? I could not understand what you want to point out."

The vast majority of mathematicians would assert that there is a set \(\mathbb N\) which models \(HA\) (or in fact \(PA\)) and that there is a unique function (which is also an injection) from \(\mathbb N\) into any model of \(HA\) (\(PA\)). I'm claiming that the 'actual' naturals are the ones that lie in this set, and that in order to make an 'actual' natural (with any degree of ease), we have to work with \(\mathbb N\) as opposed to arbitrary models of (an extension of) \(PA\) (even if not r.e and complete, we can still have elementary extensions).

"How could you do so? Of course, you can add deduction rules in order to remove axioms, but it is just an interpretation of axioms."

Points you in the direction of type theory. IIRC, MLTT0W only contains axioms for type formation of the empty and unit types, and the unique element of the unit type, and is relatively unexceptional in this regard. Mind, I'm being a bit of a pedant about the distinction between 'rule' and 'axiom' - an axiom is just a rule with no requirements. (You could do similarly for, like, anything in FOL, by having 'true' be required for a rule.)

"Is there any ambiguity? Why do you recommend so."

Well, yeah. Of course there is. Look, we may be considering any number of different theories (attempting to formalise \(\mathbb N\), \(\mathcal P(\mathbb N)\), \(L_{\omega^{CK}_\alpha}\), \(V_{I_\alpha}\), for example), and if we ever want to refer to multiple theories or multiple languages (you're addressing stuff in generalities, so do this), common sense says you specify the language. The point is for your reader to understand, so it's natural to include little phrases that prevent the reader from getting lost. An obvious one is what structure/theory/language we're working with.

"It is just a choice of terminology. I referred to the definition of a definition from the standard textbook by Kunen, who is one of the greatest mathematician. What is your reference?"

Well, we're working meta-theoretically, quantifying over definitions. You don't normally do that, so it makes sense that the term could be adapted to a more liberal context, and this is the natural extension. It still forbids undefined behaviour.

Mind, you could define \(\psi(x):\equiv(\phi(x)\land\exists!x\phi(x)\lor x=0\land\neg\exists!x\phi(x)\) instead, and implicitly do this with all your defintions (where we already know \(\neg\phi(0)\)). The details are slightly messy but ultimately trivial, and this is compatible with Kunen's defintion.

"Right. As I wrote "You are working on M itself and your definition names a natural number relative to M". (Is it the same as what you meant?) (PS: I replaced it by a clearer sentence.)"

Hmm. Yeah, I think so, but it's not especially obvious. "But I work in the universe." or "


 * In subsection 8, you use \(ZFC+Con(ZFC)\vDash(ZFC\vdash\cdots)\) where the right-hand side isn't a formula in FOL. This seems to be an error, or at least poorly phrased.


 * In subsection 10, you don't note the semantics used i.e. Henkin or full. Full obviously has its own issues, in that it needs to be defined internally to a structure with power set (which we can't generally use), whilst Henkin is essentially FOL with an extra sort and corresponding axioms.


 * You never use explicit coding. Typically, if you ever talk about encoded formulae, then you should (can?) code the formulae explicitly by using \(\ulcorder \phi \urcorner\) (\ulcorner, \urcorner) in place of \(\phi\). This is a no-loss change that makes what you're doing clearer and, given that you're trying to correct common mistakes, clarity seems desirable.