Is deductive logic consistent and how does it influence other systems?

[Canute]

Originally posted by master_coda
Of course we can prove the axioms of a non-trivial system. They're true by definition.

I'm well aware of the fact that you can't prove that the axioms are true "in reality". That's why mathematical axioms don't refer to reality. If they did, we would have to justify the axioms (we would have to show that they in fact do refer to the real reality). But since they don't we can just say "define these axioms as true" and they are.

Defining an axiom as true is not quite the sane thing as proving that they are true. Anything at all can be defined as true. Don't forget that the first axiom in a system is adopted before there is any other theorem to contradict it. You can adopt any old axiom if you take no account of 'reality'.

A system that proves its own axioms must be trivial, (contain only trivial theorems) for the same reason that all analytic propositions are trivial (in a scientific sense).

Thus

All men are mortal
Socrates is a man
Socrates is mortal

is entirely trivial unless one assumes that the first two propositions refer beyond the system to real men and the real Socrates. If they do not then the syllogism is tautological and entirely trivial.

 

[Canute]

Originally posted by master_coda
Paul Cohen originally used it to show that if ZF set theory is consistent, then ZF set theory + the negation of the axiom of choice is consistent. [/B]

Can you expand on that a bit, or give a link.

 

[master_coda]

Originally posted by Canute
Defining an axiom as true is not quite the sane thing as proving that they are true. Anything at all can be defined as true. Don't forget that the first axiom in a system is adopted before there is any other theorem to contradict it. You can adopt any old axiom if you take no account of 'reality'.

A system that proves its own axioms must be trivial, (contain only trivial theorems) for the same reason that all analytic propositions are trivial (in a scientific sense).

Thus

All men are mortal
Socrates is a man
Socrates is mortal

is entirely trivial unless one assumes that the first two propositions refer beyond the system to real men and the real Socrates. If they do not then the syllogism is tautological and entirely trivial.

Yes, you can adopt any old axiom.

Of course, if you consider any tautology to be trivial, then you must consider all of mathematics to be trivial. After all, math is really nothing but a bunch of tautologies. However, that is the strength of mathematics. We always know that the conclusions necessarily follow from the premises.

Of course there is a great deal of science that needs more than math can provide. Thus we can "extend" math and try and make it refer to reality. For example, one can attempt to apply geometry to the universe.

However, when you try to apply math to "reality", you can't change the math at will, or apply it completely beyond its scope. Or at least, if you're going to do that, don't try and pertend you are making use of math.

For exapmle, you can't change the definition of a line to make it match reality more closely, and then apply geometry. The results of geometry come from the properties of the original definition, and changing the definition changes the results.

Similarly, incompleteness applies to a specific definition of what a formal system is, and it result is about a specific definiton of completness (or lack thereof).

Originally posted by Canute
Can you expand on that a bit, or give a link.

Well, here's a page that mentions what Cohen used forcing to prove:

http://mathworld.wolfram.com/Forcing.html

And here's a page that gives a bit of an overview of forcing.

http://planetmath.org/encyclopedia/Forcing.html

 

[Canute]

master-coda

Perhaps the problem is the word 'trivial'. When I use it I don't mean 'of no importance' or 'of no utility'. I mean it in the scientific sense of 'trivially provable'.

This is synonymous with its mathematical use as (of the solutions of a set of homogenous equations) 'having zero values for all the variables' (Collin's Dictionary).

So when I say that a self-proving system is trivial I don't mean that most of mathematics is not useful, I just mean that it is trivial in a scientific sense, it makes no assertions about anything beyond the systems employed (insofar as they are tautological).

I tried your links but they're too technical for me. Is Cohen saying that for every consistent system there is an equally consistent system that can be derived from the negation of its axioms?

 

[master_coda]

The link is somewhat too technical for me as well. I only have a general idea of the method, this isn't my field of expertise.

The basic idea is that, under certain conditions, you can add an extra set to set theory without breaking the consistency of theory. The idea is this:

1. Take some model of set theory A.
2. Take some set B that is not part of set theory A.
3. Create a new set theory C that models A but also contains B.
4. If B satisifes certain conditions, we know that the model C is relativly consistent with the model A.

Cohen used this to add a new set to ZFC without breaking the consistency of ZFC. In the extended version of ZFC with his set, he was able to prove that the continuum hypothesis is false. Since his new set theory is at least as consistent as ZFC, we know that we can add the negation of the continuum hypothesis to ZFC without breaking consistency.

Cohen used a similar technique to show that we can add the negation of the axiom of choice to ZF set theory without breaking consistency.

Since Godel earlier proved that we can add the axiom of choice to ZF or the continuum hypothesis to ZFC without breaking consistency, we know that if ZF is consistent, then both ZF+axiom of choice and ZF+not axiom of choice are consistent. The same goes for ZFC and the continuum hypothesis...we can add it or its negation without breaking consistency.


When you want to apply math to reality, you have to be careful how you do it. Obviously when we start extending math to reality, we're only getting an approximate model. For example, lines drawn on a piece of paper aren't the same as lines in geometry. Lines in geometry have no width, and lines you draw do.

But the geometric lines are a very good approximation in most cases. So it's OK to apply geometry to drawings as long as we recognize that sometimes there'll be a conflict between the math and reality.

Yet you have to be careful to use at least a reasonable approximation. And when you aren't using an entire system, but only a single theorem, then you have to be even more careful, since violating even a single condition of the theorem will generally break it.

 

[Canute]

Thanks for that. Now all I need to know is what the axiom of choice and continuum hypothesis are.

I'll have a look around and see if I can find an explanation simple enough for me.

 

[selfAdjoint]

The axiom of choice says that if you have a (possibly infinite) collection of (possibly infinite) sets - any such collection - you can form a new set containing one element from each of the sets in the collection.

 

[master_coda]

The continuum hypothesis is the hypothesis that there isn't any set larger than the set of natural numbers and smaller than the set of real numbers.

 

[Canute]

Wow, I expected pages of mathematics. Thanks.

So which of these is right and wrong?

1. It been proved that ZFC is consistent whether or not these two propositions are taken as true.

2. These propositions are undecidable within ZFC, in some sense they are 'Goedel sentences'.

3. If the axiom of choice proposition is true then in ZFC there are an infinite number of possible sets.

4. If the continuum hypothesis is true then there are an infinite number of natural numbers.

5. ZFC is consistent if both propositions are taken as true, or both as false, but not if one is assumed false and the other true.

6. Neither proposition is a theorem in ZFC.

What do you reckon?

I don't get the bit about 'smaller than the set of real numbers'. (I always get confused by all these different sorts of numbers. I always forget which is what).

Cheers
Canute

 

[Hurkyl]

(p.s. ZFC means "ZF + the axiom of choice"; I think you meant to say ZF everywhere you said ZFC in this post)

1. It been proved that ZFC is consistent whether or not these two propositions are taken as true.

False; only relative consistency may be proven mathematically.

2. These propositions are undecidable within ZFC, in some sense they are 'Goedel sentences'.

The first half is correct; I'm not sure I'd call them Goedel sentences, though.

3. If the axiom of choice proposition is true then in ZFC there are an infinite number of possible sets.

Correct... but the hypothesis is irrelevant; the axiom of infinity in ZF guarantees an infinite number of sets.

4. If the continuum hypothesis is true then there are an infinite number of natural numbers.

Again, this is guaranteed by the axiom of infinity.

5. ZFC is consistent if both propositions are taken as true, or both as false, but not if one is assumed false and the other true.

I *think* that ZF + any combinations of accepting or denying the axiom of choice and continuum hypothesis is relatively consistent to ZF.

6. Neither proposition is a theorem in ZFC.

Because the axiom of choice is an axiom of ZFC, it is also theorem in ZFC, via the trivial proof:

The axiom of choice.
Therefore, the axiom of choice.

However, it is correct that the axiom of choice is not a theorem in ZF.

 

[Canute]

Whoops. yes I meant ZF.

Could you just explain whatyou mean by relative consistency?

 

[master_coda]

Originally posted by Canute
Could you just explain whatyou mean by relative consistency?

Relative consistency means just means that if one system is consistent, then the other is as well.

For example, if ZF is consistent then ZFC is consistent.


Of course if ZF were to turn out to be inconsistent then relative consistency doesn't help us. But it weakens the argument of someone who accepts the ZF axioms but disagrees with the axiom of choice.

 

[phoenixthoth]

The axiom of choice says that if you have a (possibly infinite) collection of (possibly infinite) sets - any such collection - you can form a new set containing one element from each of the sets in the collection.

sometimes people specify those sets must be all nonempty but that is implied, i think.

 

[Canute]

Originally posted by master_coda
Relative consistency means just means that if one system is consistent, then the other is as well.[/B]

Got it thanks.

Presumably this is true for every system, mathematical or not, since the consistency of one system can only be checked by using another system.

 

[master_coda]

Originally posted by Canute
Got it thanks.

Presumably this is true for every system, mathematical or not, since the consistency of one system can only be checked by using another system.

Well, pretty much. We don't know if deductive logic is consistent, so anything that uses it is kind of stuck.