Circularity in defining formal language

Conversation started Jun 23, 2012 at 16:23.

Jonas Teuwen

Jun 23, 2012 16:23

Hmm, can someone explain me this: If I want to define sets I need a language right and some logical scheme of deduction.

But I also need variables otherwise I cannot say anything can I?

But... so you assume a set of variables. But you did not define a set yet.

N3buchadnezzar

Ahiy thar @JonasTeuwen boy =)

Martin Sleziak

I wonder whether someone already has asked this on main site.

Rajesh D

@Jonas : Hi

I guess no need for variables to define a set. I may be wrong.

Martin Sleziak

I found this

When I read it, I have feeling that I do not understand metamathematics at all.

If I tried to answer your question @Jonas, I would say that there are two levels: 1) Logic in the sense how do we think 2) our attempt to formalize the logic using theories, first order languages and stuff

N3buchadnezzar

@MartinSleziak It is always like that. Looking at mathematics you should encounter in two-three years always looks greek.

Jonas Teuwen

Jun 23, 2012 16:29

So your set of variables is like in a metalanguage?

Martin Sleziak

So we start with natural logic.

Jonas Teuwen

Aha.

Yea, I see.

You describe the elements in a metalanguage.

And then you do logic with that.

And if you want to describe the metalanguage you need another one...

Holy cow! Infinite! Blows up!

Martin Sleziak

:-)

J. M.

@JonasTeuwen Turtles again... :)

Martin Sleziak

This seems related: p.45 in Just-Weese, the paragraph which starts: Isn't Definition 3 circular? That is, aren't we defining natural numbers in terms of natural numbers?

Although it is about definition of natural numbers, but kind of about similar circularity.

Rajesh D

Jun 23, 2012 16:33

@Jonas : How did you happen to run into this?

Jonas Teuwen

Just thinking about things.

Martin Sleziak

@Jonas I believe these questions at MO are closely related to your question: The sets in mathematical logic, Breaking the circularity in the definition of N, Don’t the axioms of set theory implicitly assume numbers?.

In particular, Stefan Geschke's answer seems nice.

Jonas Teuwen

@MartinSleziak Thanks this is great stuff. I will go to a friend to have dinner now. Check it out later.

Martin Sleziak