What is the main problem

why we can't reduce all mathematics to logic?

>What is KFC.
On the other hand this doesn't mean we can solve all problem if I have correctly understood what Goedel means.

Then you didn't understood anything about Godel

That's not what the Incompleteness Theorems say.

Logic is a sub-field of math you retard

Well he and Whitehead were going to do a second volume on geometry, but they stopped first because of "intellectual exhaustion". Then before they got started again Godel published his incompleteness theorems, and they both realised that it could never be completed. So they stopped permanently.

Well, you can spend a few words and explain in brainlet terms what Goedel proved.

...

Formal logic is just a kind of mathematics, so its not truly getting reduced to something outside of math. If the basic rules and concepts of mathematics, that cant be broken down any further are mathematical (which they are), then you cant reduce mathematics to something exogenous to mathematics.

I think another more philosophical point about the theorems, is that "provable" and "true" arent actually independent concepts. There are no provable false propositions, for example. If its provable, its true. So the whole spookiness of Godel's theorems rest in the whole "true but unprovable" part. Maybe I just havent read Godels theorems close enough, but I dont believe that anything is true but unprovable. Mathematics isnt a field of unknown information. You have all the rules and principles. Saying it could be true, but unprovable, is like saying the rules and princples have nothing to say about the true proposition that you want to prove, which is clearly false. The rules of arithmetic always bear meaning on all arithmetic propositions. Theres no proposition of arithmetic that reflexivity doesnt speak to, for example. To say its unprovable true, is to say its not derived from the fundamental rules of math, and thats just pure nonsense. If its not derived from the fundamental rules, its just not a part of that domain of math in the first place.

Furthermore, I dont think any mathematician could offer a strict and complete definition of "provable". Proof theory simplifies it so a chain of logical inferences, but I think many proofs are a lot looser to that, and the chain of inferences is only obvious after the fact. Pictoral proofs are proofs, but they dont depict or contain explicit propositions.

Looks like YOU are the one that pretends to understand his theorem.

We can't prove every theorem. I'll explain more if your brainlet body wants to impress people at the next math party you go to.

We should teach logic gates in middle school

Axioms are true, but unprovable.

For any math system you invent, there will be an infinite number of axioms. You can never write down all the axioms, therefore there will be infinite true statements that you cannot prove.

You can only prove something if it comes from the fundamental axioms. Every proof will start with axioms make steps to a conclusion. You can't prove an axiom, because the proof for it would require for you to suppose that that axiom is already true!

> Axioms are true, but unprovable.
>You can't prove an axiom, because the proof for it would require for you to suppose that that axiom is already true!

How are they unprovable? Its a very simple proof.

Suppose p

then p

QED.

You could say I am just assuming the axiom, and that I could assume any axiom I want and it would be true. You would be right but there is nothing scary here, because if I assume something crazy, then I am doing a different kind of mathematics that has no bearing on the non-crazy parts. If I assumed something opposite of an axiom of arithmetic, its not that I am making any proposition in arithmetic true and false, its just I am doing something OTHER than arithmetic.

>You can never write down all the axioms, therefore there will be infinite true statements that you cannot prove.

Thats a really poor understanding of Godels theorem. Godel wasnt saying there are so many true things, that one doesnt have the life span or writing capacity to record them all on a piece of paper. He wrote proved theorems, which say something mathematical.

am I understanding Godel correctly in that every mathematical system has some underlying assumption which cannot be disproven based on the rules of that system. (?)

I like that, because it means inherently mathematics is always constrained and can never be consistent within its own system (which would imply a logic 'god', or some kind of self-reinforcing logic strucutre that resembles AI)

Neat

> in that every mathematical system has some underlying assumption which cannot be disproven based on the rules of that system. (?)

No. If P is your principle, the theorems dont say "There exists no Q such that Q -> P". Its saying "Q is true, while no p of P Principles -> Q".

That is not a proof you literal moron.

Watch me prove you are a faggot.

p = "you are a faggot"

Suppose you are a faggot

Therefore, you are a faggot.

>Godel wasnt saying there are so many true things, that one doesnt have the life span or writing capacity
That is exactly what he said you popsci retard. That's not the most important part of the theorems, but he did prove there are infinite axioms for any system. Please try harder, get some help.

>Therefore, you are a faggot.

Well I cant argue with that. But all youve established is that thats true within a system, but what that system has to say about me and whether I am a faggot is still an open question.

>That is exactly what he said you popsci retard.

omg no. You are the popsci retard if you think that.

We can. The computer is a living demonstration of Russell's hypothesis that all mathematics can be reduced to logic. Any mathematics which can be represented in a program on a computer (e.g. Wolfram Mathematica, etc) can alternatively be represented as a collection of logic gates. And in fact *is* being represented as a collection of logic gates by the computer.
You can build a functionally complete logical system out of only NAND gates (or only NOR gates). All mathematics is reducible to a collection of NAND operations.

Russel is a cuckold and spent his life enjoying giving sexual pleasures to women FOR FREE and he was claiming to be some rationalist and empiricist at the same time, like any other secular humanist has been for the last few centuries.

truth does not exist in logic nor math, at best you have a validity of a derivation of a statement, from some statements, according to some rules that you read in a book.

>Axioms can't be proved
Reductio ad absurdum?

>truth does not exist in logic nor math, at best you have a validity of a derivation of a statement, from some statements, according to some rules that you read in a book.

Are you correcting me or working with me here?

Mathematical truths are truth on the basis of mathematical rules. Whether Im a faggot is truth if I am actually a faggot. But even the english language is a system of rules like the fields of mathematics. You can conceive of a way of talking, and how that way works, independently of how it corresponds to the real world. So, if Im a faggot, its true for that reason only, but, if we have a system that exists for the purpose of declaring whether I am a faggot, the rules of that system are true or false regardless as to whether I am actually a faggot.

Any logical system is set up to avoid the necessity of proving axioms or postulates. A logical system is a closed system in which axioms and operations define the total extent of any consistency (or truth).

Axioms aren't truth claims, theyre foundations of truth claims

this is fucking backwards you Luddite. Numbers are real-world variables that are only stand-ins for the universal nature of logic.

You can reduce all of mathematics to logic. The problem is it is inherently inconsistent.

Wrong.

how so?

Isn't a proof more like this:
Assume P. Then show that P -> Q is a tautology when a set A of assumptions are true. Then:
(P and A) -> Q

It isn't "true" in the normal sense, rather it is true for all instances of P and A being true. So if p="you are a faggot" is true then certainly p is true, if the assumption is true in said system.

It is not the case that the system must be inconsistent.