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

[Canute]

Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

 

[phoenixthoth]

i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

hmm... what are we really talking about here? read that one more time and you'll get the allusion.

 

[master_coda]

Originally posted by Canute
Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

What outside of the system does that axiom refer to?

 

[phoenixthoth]

math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

 

[master_coda]

Originally posted by phoenixthoth
math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

You cited a number of examples of axioms that "correspond" to what we see the real world. But it is in no way necessary for axioms to manifest themselves some way in reality.

For example, non-Euclidean geometry generally replaces the parallel lines axiom with an axiom that is contradictory to the parallel lines axioms. And non-Euclidean geometry is just as much of a valid system as Euclidean geometry.

Most of the other examples are very circular...they're concepts that relate to the real world because humans use them...and we use them because they relate to the real world. Perhaps they only seem to be "intrisically true" to us because we were raised to believe that.


Whether or not the axioms are "true" outside the system is irrelevant. It's usually best to start with axioms that intuitively seem to be true, but we can just as easily do the opposite, and the math is just as valid.


Of course, using the mathematical definitions of what a formal system is and what axioms are tends to rob those concepts of any philosophical meaning...but if you want to use the incompleteness theorems then you have to use the mathematical definitions. You can't just discard those definitions because you prefer more meaningful ones and keep the theorems at the same time.

 

[phoenixthoth]

part of what i said earlier was:
an axiom only refers to the system it is part of the foundation of (this is what you said in more words). that's what it refers to but that is [edit: usually] not why it exists, ie, why it was created.

as you said, the new axioms from post 1850 or so had no correspondance to reality except that what we imagine is real to an extent in that we really imagine. so those axioms still corresponded to or were invented by an observation of an aspect of reality: the reality inside our minds which of course is not real in the usual sense. but then it turned out that certain axiomatic adjustments lead to reality anyway, which is kind of interesting. i wonder what more we can piece together by adjusting the way we think.

this leads back to max tegmark's TOE article where existence is equivalent to freedom from contradiction.

 

[Canute]

But a non-trivial axiomatic system cannot prove its own axioms. They are only partly part of the system.

(On the problem of explaining the cosmos) – “Every proof must proceed from premisses; the proof as such, that is to say the derivation from the premisses, can therefore never finally prove the truth of any conclusion, but only show that the conclusion must be true provided the premisses are true. If we were to demand that the premisses should be proved in their turn, the question of truth would only be shifted back by another step to a new set of premisses, and so on to infinity.”

Karl Popper – The Problem of Induction (1953, 1974) from ( http://www.dieoff.org/page126.htm)

Aristotle got around this by saying that there are earlier premisses which are indubitably true, and which do not need any proof. He called these ‘basic premisses’.

For both Aristotle and Pooper the problem was the in principle impossibility of proving ones axioms, or basic premisses, without rendering the system trivial.

Inasmuch as human knowledge is based on axiomatic reasoning and proofs Popper also writes this:

“What we should do, I suggest, is give up the idea of ultimate sources of knowledge. And admit that all human knowledge is human: that it is mixed with our errors, our prejudices, our dreams, and our hopes: that all we can do is grope for truth evn though it be beyond our reach. We may admit that our groping is often inspired, but we must be on our guard against the belief, however deeply felt, that our inspiration carries any authority, divine or otherwise. If we thus admit that there is no authority beyond the reach of criticism to be found within the whole province of our knowledge, however far it may have penetrated into the unknown, then we can retain, without danger, the idea that truth is beyond human authority. And we must retain it. For without this idea there can be no objective standards of enquiry; no criticism of our conjectures; no groping for the unknown; no quest for knowledge.”

Karl Popper – ibid.

I believe he was right given his scientific definition of knowledge, but wrong because he defined knowledge too narrowly.

Descartes recognised the problem, and thus chose a fundamental axiom that was in principle impossible to prove.

 

[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.

 

[phoenixthoth]

one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

 

[master_coda]

Originally posted by phoenixthoth
one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

I'm not very familiar with the details of forcing. But it isn't for proving consistency, it's for proving relative consistency.

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.

 

[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.