A problem with the Axiomatic Foundations of Mathematics

[Studiot]

A couple of references for you.

The Open University set book on this stuff is rigourous but not too technical.

Computability and Logic
by Boolos and Jeffrey

Cambridge University Press

Forever Undecided a Puzzle Guide to Godel
by Smullyan

Oxford University Press

is more populist but well written by someone who knows what he is talking about.
This author also has some more formal texts in logic.

go well

 

[Skrew]

What does being "based on axioms" have to do with anything?

If you left the real numbers as being "fuzzy" instead of representing them as an axiomatic system, it seems you would be able to more easily cope with potential contradictions by ruling out certain results on grounds of "logical absurdity".

Geometry exists independently of it's axiomatization, just as the integers and their properties exist independently of the Peano axioms. It wouldn't be geometry that would be shown inconsistent, it would the the particular axiomatization, just it wouldn't be the integers and their properties that would be shown to be inconsistent, it would be the Peano axioms.

Then what are integers? Can you define integers to be as they are viewed at the intuitive level? Can you construct a theory based not on specific axioms but on our intuitive understanding of integers? Maybe this ties in with my reply to Hurkyl

I posted a bunch of links on the first page of the present thread. Mine is the second post.

https://www.physicsforums.com/showthread.php?t=511309

Were any of those helpful?

On the other hand, most set theorists believe PA has actually been proved consistent. It's something specialists can argue about. I believe if you look at the Mathoverflow link I gave, it has some references.

The two famous mathematicians who think PA may be inconsistent are Ed Nelson, who many people don't take seriously; and Vladimir Vovoedsky, who has a Fields medal for his work in algebraic topology, so people have to take him seriously. Although a lot of people don't take seriously his claim that PA might be inconsistent.

The links I gave have a lot of back-and-forth about all of this.

Could you provide more information on regards to PA being proved consistent? I did some looking online I found a wiki article about Gentzen's consistency proof and other proofs using logic but I want more.

In particular can you recommend a book which covers these topics?

A couple of references for you.

The Open University set book on this stuff is rigourous but not too technical.

Computability and Logic
by Boolos and Jeffrey

Cambridge University Press

Forever Undecided a Puzzle Guide to Godel
by Smullyan

Oxford University Press

is more populist but well written by someone who knows what he is talking about.
This author also has some more formal texts in logic.

go well

Thanks, do you know if these books cover Gödel's incompleteness theorem and the consistency proofs of PA/real number arithmetic?

If not I would be quite happy to receive suggestions on more books.


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Also a question I have never found the answer to is, how does the chain of consistency work? Supposing PA is consistent does it follow that the axioms for the real numbers are consistent? Or do you need additional things other then PA in a construction of the real numbers?

 

[Hurkyl]

If you left the real numbers as being "fuzzy" instead of representing them as an axiomatic system, it seems you would be able to more easily cope with potential contradictions by ruling out certain results on grounds of "logical absurdity".

Then math was finished centuries ago and all that's left is engineering. All of the easy theorems have been proven, and anything left is too complex for us to have any confidence in.

IIRC, the main reason for the interest in foundations is because of the development of real analysis a couple hundred years back progressed beyond the point where "fuzzy" foundations were good enough.

Also, there are practical matter -- clear foundations often pave the way to progress in the subject. And are often simply easier to work with than a purely intuitive formulation.

 

[TylerH]

Then what are integers?

They are what they are, intuitively. They're primitive notions.

 

[Stephen Tashi]

They are what they are, intuitively. They're primitive notions.

The link you gave doesn't say the integers are primitive notions and I don't think you'll find any authoritative sources that say they are.

 

[lostcauses10x]

After looking at the axiomatic systems of modern mathematics and asking myself what proves they are self consistent I went looking for an explanation and so far I have found only that they have not been proven self consistent nor likely will a proof ever exist. So the possibility of a contradiction being present within the system exists. Therefore when using the system you assume it has no contradictions present within it.

I find this incredibly disturbing as it means every proof I have written would become worthless should the axiomatic system it is written in be demonstrated to be inconsistent.

I find this so disturbing that I question if I want to pursue my studies in mathematics, one thing I always liked about mathematics is that I considered it built on unshakable ground but this appears not to be the case.

Has anyone else experienced this revelation? How do you deal with it?

Man is not an absolute. We build in our own limits and mistakes.
Your ability to see this situation is why you should stick with it.
Others have advised you to look into formal logic etc.
I would add a good lesson or book on history of the subject.
There also exist papers wrote out there covering the same questions and more by noted mathematicians.
Finding of such may also be of value to you and your decisions.