Logic - 2017-09-28

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4:14 AM

@user21820 Ive been meaning to ask, how have you resolved the problem you had with ZFC a while ago?

I don't remember the technical terms, so in the next sentence I substitute "reason" for the term I have forgotten. Please don't ascribe to it the technical meaning.

I think it went along the lines of "anything ZFC can reason about, it can prove consistent, yet there exists a model that it can reason about which can prove itself inconsistent."

As a word of warning, I may have dreamed up this conversation, since I can't seem to find it using the search feature.

user21820

@user400188 Hello! I've talked to quite a number of people about ZFC, so I'm not sure what it was we were talking about the last time. There is a reasonable way of interpreting what you "dreamed up", that is consistent with what I must have said before. I can easily explain it again. But it has nothing to do with my philosophical objection to ZFC, since it applies to any reasonable foundational system anyway. =)

By the way, the search feature is quite broken, so don't rely on it.

in This is the Realm of Simply Beautiful Art, yesterday, by user21820

@SimplyBeautifulArt @MartinSleziak: When I was looking for my message that I just linked to, I found that there seems to be something wrong with the search function. I tried searching for "ordinals" but the search results only reach Jun...

in This is the Realm of Simply Beautiful Art, yesterday, by Mithrandir

Yeah, chat search is broken at the moment.

user400188

Thats a shame, a lot of my favorite conversations on the internet are buried somewhere in this (and Simply's room.)

user21820

Well, you can try Google

Oops Google says this site's robots.txt doesn't allow indexing...

By the way, you can save interesting conversations that will always be easily accessible to you.

To do so, click on "room▼" on the right and then "create new bookmark".

user400188

Oh thats a nice feature.

user21820

The only downside is that all bookmarks by all users will appear here, and even room owners don't have the right to remove them...

Not like I would need to remove your bookmarks though.

So bookmark away!

user400188

4:25 AM

I'm glad the 0.999..=1 thing is already there. It was one of the explanations I would have saved.

user21820

That's nice. Unfortunately someone doesn't like the contents of my answer. No idea who though.

It mainly happens with my philosophical posts, but also happens with some purely mathematical posts that don't follow conventional methods. Can't be helped haha..

Anyway, back to the point about ZFC, any reasonable meta-system proves that if PA is consistent then PA+¬Con(PA) is consistent. ZFC not only proves that but also proves PA has a model and hence is consistent.

user400188

how should I read the + symbol?

user21820

PA+¬Con(PA) is a short-hand for PA with the additional axiom ¬Con(PA).

So you have the same language, same logic, same axioms plus just one more.

user400188

ah ok. I was thinking along those lines but I wanted to confirm the assumption.

user21820

¬Con(PA) is some arithmetical sentence that we can explicitly write down. It does not have meaning until we evaluate it in the standard model, namely the natural numbers in our meta-system.

For more details you can see the following:

A: How could a statement be true without proof?

Your confusing stems from the way many articles about Godel's incompleteness theorems are extremely imprecise. Here is a proper definition. $\def\nn{\mathbb{N}}$ We say that a sentence $φ$ over a language $L$ is true in an $L$-structure $M$ iff $M \vDash φ$. For convenience, when $L$ is ...

user400188

4:33 AM

Sorry about this, but i appear to have mismanaged my time. I'm at uni at the moment and I am about to start a class.

Thats for the explanation so far though. Ill probably be back in about an hour or so.

user21820

Sure sure. I'll be around later today.

Anyway here are two other related posts:

A: True and unprovable vs. unprovable and unrefutable

See this post for a precise generalization of the incompleteness theorems. $ \def\nn{\mathbb{N}} $ There I have defined an arithmetical sentence over a formal system $S$ that interprets arithmetic to be of the form $ι(φ)$ for some sentence $φ$ over PA, where $ι$ is the translation that witnesses ...

This one is to cater to systems that don't actually use the language of arithmetic, so you need some (computable) translation.

A: Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

The problem is with your notion of logic and truth. Before you even can talk about a sentence, you need to specify your language, and before you can talk about whether it is provable or disprovable (or neither), you need to specify the formal system (which must use the same language or larger), a...

This one explains what logicians mean when they say some mathematical statement is equivalent to some arithmetical statement.

@LeakyNun: I just found an old post where I explained some stuff about the derivability conditions and why consistency hardly implies justifiability of a formal system:

A: Proving the existence of a proof without actually giving a proof

There are two different notions of proof that other answers did not distinguish. Proofs from 'without' One is when we are in a meta-system and talking about proofs inside a formal system. For example, the completeness theorem for first-order logic says that for any set $S$ of formulae and formu...

9 hours later…

user21820

1:30 PM

@amWhy: Hello! How have you been? =)

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