Thread: Why providing a model means a theory is consistent?

Lately I lose sleep at night worrying about the consistency of various theories. Linear algebra, number theory, set theory etc.

Question 1: How do we know things like Russell's paradox (which ultimately lead to reformulation of axiomatic set theory) won't repeat themselves in these theories?

Many times you take a problem and look at it from a different direction, reaching a conclusion that is consistent with what you already know. Sometimes it seems like magic.

A consistent theory is one that you cannot prove a claim and its opposite using its axioms. (reaching a contradiction)

Question 2: It is said that when a model is found for a certain theory, that theory is consistent. Why? Why can't it be that our model is inconsistent?
Case in point, non euclidian geometries - not agreeing with the 5th postulate, how do we know these are consistent. Perhaps I can construct something using such a geometry that would contradict itself, maybe this doesn't prove the 5th postulate doens't follow from the first 4.


Any thoughts?

Thanks,
Leo

In regards to your Question Number 1, I would say that Zermelo-Fraenkel set theory solves Russell's paradox by preventing its statement. In ZF set theory, you aren't allowed to talk about the kinds of sets Russell talked about. See this book for a good intro to ZF set theory. ZF set theory is generally regarded as a sufficient set theory for the disciplines you mentioned, although most authors these days, for brevity, just use naive set theory.

Question Number 2 is for a better logician than I. I'm sure emakarov could weigh in, or Moeblee, or others.

A few thoughts, I guess...

1) Try sleeping pills.
2) Everything that is built is built on a foundation and no foundation encompasses the whole of anything.
3) Many things are sufficiently reliable that we can build things that don't fall down. Don't discard them because something we can't build might fall down.
4) If newer and fancier stuff comes along that tells us we've been wrong for centuries, so what? We're only dealing with validity in a give framework, not truth.

Originally Posted by TKHunny

A few thoughts, I guess...

1) Try sleeping pills.
2) Everything that is built is built on a foundation and no foundation encompasses the whole of anything.
3) Many things are sufficiently relaible that we can build things that don't fall down. Don't discard them because something we can't build might fall down.
4) If newer and fancier stuff comes along that tells us we've been wrong for centuries, so what? We're only dealing with validity in a give framework, not truth.

Practical thinking may help me sleep a bit. But still, the possibility of one day constructing something contradictory (a plane that falls down after we've built a prototype worth 1 billion dollars) should scare us.
Let's agree to keep this discussion theoretical

What's not theoretical about proceeding BEFORE we know EVERYTHING? It is foolishness to wait for all knowledge. Nothing ever would occur. If we are motivated by fear, where would we ever get a pioneer?

Originally Posted by leolol

Question 2: It is said that when a model is found for a certain theory, that theory is consistent. Why? Why can't it be that our model is inconsistent?
Case in point, non euclidian geometries - not agreeing with the 5th postulate, how do we know these are consistent. Perhaps I can construct something using such a geometry that would contradict itself, maybe this doesn't prove the 5th postulate doens't follow from the first 4.

You are right; building a Euclidean model of a non-Euclidean geometry only proves that the latter is consistent relatively to the former, namely, that if the Euclidean geometry is consistent, then so is the non-Euclidean geometry.

Have any efforts been made to prove the Euclidean theory, linear algebra etc. are indeed consistent?

Originally Posted by leolol

It is said that when a model is found for a certain theory, that theory is consistent. Why?

The notion of "an inconsistent model" doesn't even make sense. What is consistent or not consistent is a set of formulas (a theory being a certain kind of set of formulas). A set of formulas G is consistent if and only if there is not formula P such that G proves both P and ~P (a contradiction). So it doesn't even make sense to say "a model is consistent" or "a model is inconsistent".

If G has a model M then G is consistent as follows:

A model M determines, for every sentence S in the language of G, that S is true in M or S is false in M, but never is a sentence both true in M and false in M (that fact follows trivially from our technical (or even informal) definition of 'true in the model').

M is a model of G, so if G proves a sentence S, then S is true in M, and since no contradiction is true in any model, we have that G proves no contradictions.