/sci/ - Science & Math

>>10581286
>I just want to see my wife again. Let me go, please. I beg you.

>>10581286
>Spare me, I no longer care about my wife, please, just let me go.

>>10581286
Does [math]2^{\aleph_0}[/math] equal [math]\aleph_1[/math], yes or no? Should be an easy question for a big strong setfag such as yourself

>>10581512
unknowable according to ZFC. Dosent mean it dosent have a truth value.

>>10581564
Then you should be able to construct a rational way to decide the correct truth values of statements. https://en.m.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems oh...

>>10581608
Hey there! I see you're new to discrete math. You seemed to have implied that incompleteness implies formalism! Whoops! It was even godel himself that viewed his own theory as proof that human cognition is not mechanical in nature. In other words some problems like the halting problem can be decided by ideal human minds! It's crazy! I would recommend trying this subject out, it's a blast!

>>10581643
Everything is mechanical.
There is no correct algorithm for the halting problem, but heuristics can approach correctness. Human brains are heuristic machines.

>>10581643
even turing wasnt as much of a faggot as you are

>>10581286
you appear to be diseased with cancer and leprosy at the same time

>>10581512
(1) Platonism doesnt claim that any definable feature of a mathematical object must be determinate. This is generally associated with logicism, not Platonism (since the assumtpion then, is that the rule of logic combined with the definitions of objects is sufficient to determine whether they exist and what their properties are).

(2) In fact both the first incompleteness theorem and Godel's contribution to the continuum hypothesis were originally argued by Godel to provide support for Platonism again logicism (and the second inconpleteness theorem was similarly seen as posing a challenge to formalism).

>>10581608
As another anon mentioned, you seem to be completely misunderstanding the incompleteness thereoms. In fact the whole point of the first incompleteness theorem is to show that in any theory that axiomatizes first order Peano Arithmetic there exist statements that MUST BE TRUE under the semantic interpretation of the model, but are UNPROVABLE under the available axiomatization. In other words, the whole point of Godel's theorem is that it should that there are sentences that must be true even if we cant tell that they are.

>>10581707
>under the semantic interpretation of the model
youre supposed to work IN the model tho

>>10581286
Because you don't live in reality.

>>10581759
>youre supposed to work in the model

Yes anon, that was my whole point. The first incompleteness theorem shows that inside the model, the sentence cannot be proven nor can its negation. The problem here is that you cant prove the sentence based on the axioms that are available to you within the model even though your semantic interpretation entails that it must be true.

If the statement were simply not provable (without also being true at the same time), then the system would not necessarily be incomplete, but would rather meeet the slightly different condition of being "undecideable". In fact, another pair of classical results in foundations is that first order logic is undecideable (Church's undecideability theorem) since there is no algorithm that can decide for an arbitrary sentence of first order logic whether it is provable or not (similar to Godels incompleteness theorem). However there is another result due to Godel called the completeness theorem for first order logic, which states that first order logic is complete, meaning that any true formula must be deducible from the axioms (even though Church's undecidibility theorem tells us that we have no way of determining how to actually show that every true sentence is deducible).

>>10581643
Reading lots of Dr. Zeilberger’s opinion blog, now aren’t we?

>>10581286
>Please I want to feel like a person again, I look in the mirror and I don't recognize the person staring back at me.

>>10581286
>I don't care about physical attraction or sex any more, I just want to be loved unconditionally.