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/sci/ - Science & Math

Please explain, in terms a brainlet can understand, the implications of Gödel’s incompleteness theorem.
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>>10468204
we just don't know
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for any given set of axioms, there are true statements which cannot be proven using only those axioms
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>>10468211
This, but also:
If a given set of axiom purports to prove any statement, then the set of axioms must be inconsistent.

(absolute brainlet here btw, but i think thats right)
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>>10468259
i think you are confusing this with the second part of theorem, which states that a set of axioms cannot prove it's own consistency. consistent systems of axioms still exist, even if they cannot prove EVERY statement.
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>>10468204
this >>10468211 but also, if you want to extrapolate it to the final conclusion, it implies there exists an infinite set of axioms, and that math is infinite.
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what is an axiom?
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>>10468282
a statement which we assume to be true, without having to prove it. a self-evident truth.
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>>10468204
He showed even math isn't 100% foolproof using something like "This statement is false" but with math.
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>>10468204
In any system of reasoning there are true things we can't prove
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>>10468292
That's baby's first definition of an axiom, it's really just about coming up with consistent and interesting systems.
>>10468531
Not really about math but about formal systems, and that's for each individual one. We can come up with different axiomatizations for the same theories. They can even have a completely different foundation.
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>>10468204
guys guys guys. You are all so very fucking wrong. There are complete axiomatizations, obviously. Jus take all the true arithemtic statements on the natural numbers: boom you just got a (worthless) complete and consistent axiomatization of arithmetic.

Why is it worthless? Because generating this axiomatization means writing down every true statement of arithmetic.

So let's look at axiomatizations that one can somehow generate with an easy, algorithmic process, i.e.\ such that in principle one can write a program that gives all the axioms of your system. Now does Godel guarantee there are sentences which cannot be proven in this system?

Not quite! You see, we can still take a relatively small system (= 'portion of maths'), and try to axiomatize it in an algorithmic way. If the stuff you try to axiomatize is not too complicated, you can do this, e.g.\ the theory of dense linear orders.

So when can we Godeljerk on our theory then? Well, if your system is axiomatizable in an algorithmic way _and_ you can express a big enough portion of arithmetic, then there is a sentence in your language which your system cannot prove or disprove, save when your system is inconsistent.
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>>10468578
what is defined as a "big enough portion of artihmetic"? As in, how would you know where the cut-off is for a system to become incomplete?
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>>10468583
Well, you go to a proof of Godels theorem and you look at the exact assumptions on the given system. But I know that having Peano Arithmetic is enough to guarantee incompleteness, but that's not the minimum. On the other hand, Presburger arithmetic (addition and induction) is complete. So the cut-off point is somewhere in between.
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>>10468204
You cannot prove all mathematical truths. There are mathematical claims that are true, but cannot be proven. And you cannot make a computer program that finds out for any mathematical claim whether or not it is true.
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>>10468586
I see now. thanks for clearing that up!
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>>10468204
Godel’s Theorem simply says that not all true things are provable.
Axioms are "tools" and mathematicians don't really care about them. They are like a microscope is a "tool". Different "tools" will be used, given the usefulness of any given "tool", and they are only useful insofar as to study an actual "thing" which is true.
That "thing" being studied is the natural numbers.
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>>10468594
Does that mean that there are somethings in this world that are true but can never be proven?
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>>10468615
correct.
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>>10468204
You can guarantee to be always consistent or always correct, but not both at the same time all the time.
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>>10468204
>tfw no cute paranoid schizo-physicist bf
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>>10468278
This.
There exists one set of true mathematical axioms that are infinite in cardinality. But it's still just one set. ZFC is part of it and it's true, but that's only 9 axioms of an infinite set of them.
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Thank you.
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>>10468259
>If a given set of axiom purports to prove any statement, then the set of axioms must be inconsistent.
What? No!
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>>10468204
You can never set axioms so that they are all deductive from each other. In other words, you can never explain everything using only logic.

In his god proof, he extrapolates that to our universe. The universe can never be logically explained by itself. However, it exists. Therefore, there must be something beyond the universe (and beyond logic). That something is God.
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>>10469339
>>10468266
guys, *any* statement means some statement P and also Not P, since Not P is just anoter statement.
If a set of axioms can prove *any* statement it is inherently inconsistent
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>>10469347
Yes but don't constructivists deny the law of excluded middle?
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>>10468211
I never liked this explanation. To someone uninitiated, it seems like it presupposes that there is an objective truth value to every statement. It's confusing, because it doesn't seem obvious a priori whether something like the Continuum Hypothesis should be true or not.

IMO, a better way to explain it:
"For certain sets of axioms (recursively enumerable, first order systems capable of modelling the natural numbers) there exists a statement that is true in some models and false in some (if the system is consistent)".

Just to give some intuition, here's a dumb example of what it means for a statement to be "true in some models and false in some". Note that it's not actually connected to Gödel.

Given the axioms of a ring(https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition), does there exist an element x such that x^2=-1? The answer is clearly, it depends. In the complex numbers it's true, but it's false in the reals, for example.

So we have at least one model where the statement is true, and at least one where the statement is false. So clearly, the statement is not provable from the axioms of a ring.

What Gödel's theorem says is that for "sufficiently powerful axioms" there are always, inevitable, statements like that (assuming consistency).

An inconsistent system, on the other hand, is a system capable of both proving and disproving (simultaneously) every statement you can possibly form. So it's completely useless.

I also want to mention Gödel's completeness theorem, another theorem of Gödel which is just as important but not mentioned as much. What Gödel's completeness theorem states is that the statements provable from a set of axioms are PRECISELY the statements that are true in EVERY model.
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>>10468204
Implications are pretty useless, like you can construct useless propositions like "this statement can't be proved".
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>>10469517
Go ahead and write that statement in formal logic then.
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>>10468615
Yes, like God
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>>10469522
There's no point to do useless things.
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You're not done until the final statement titles it. Not with certainty. It's otherwise a matter of collusion. Like that black judge and ocasio cortez fucking on cam for ppl to "indulge" his a rimony as just lovers righta. The nudes were on b.

She tiyled it with fucking and reading the guy the law like they were Miranda rights.

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