∞ is a free variable

[wtf]

It's a trivial construction in sheaf theory to resolve this paradox and allow both the definite and indefinite parities to co-exist.

LOL.

1) Without looking it up, define a sheaf and give several examples.

2) Without looking it up, provide a brief overview of sheaf theory and its use in various branches of mathematics.

3) Give the "trivial construction" you reference.

Queue some retort about how I am using terminology I don't understand.

You make it too easy.

[Skepdick]

It's a trivial construction in sheaf theory to resolve this paradox and allow both the definite and indefinite parities to co-exist.

LOL.

1) Without looking it up, define a sheaf and give several examples.

2) Without looking it up, provide a brief overview of sheaf theory and its use in various branches of mathematics.

3) Give the "trivial construction" you reference.

Queue some retort about how I am using terminology I don't understand.

You make it too easy.

Why do I have to define it if I can just go ahead and use it...

Let X be the topological space of integers, with the discrete topology.
Define a sheaf F on X where for each open set U in X: F(U) = {functions f: U → {odd, even, undetermined}}
For any "standard" integer n, the section over {n} maps n to either 'odd' or 'even' preserving our ontological intuitions.
For TREE(3), the section over {TREE(3)} (extend this section as needed) maps TREE(3) to 'undetermined' preserving our epistemic intuitions.
The global section g ∈ F(X) is defined as:
g(n) = 'odd' if n is odd
g(n) = 'even' if n is even
g(TREE(3)) = 'undetermined'

Define a cover C on X: C = { {n} | n is an integer } ∪ { {TREE(3)} }

blah... blah ... blah ... blah...

Anybody who's got a gram of clue about distributed systems design has done some form or shape of this exercise when trying to glue together local properties into something unifiable at global scale.

It's just systems integration.

[wtf]

For TREE(3), the section over {TREE(3)} (extend this section as needed) maps TREE(3) to 'undetermined' preserving our epistemic intuitions.

This is what they call assuming that which is to be proved.

You seem to believe TREE(3) has no determinate parity. That's wrong, if not flat out insane.

But even if it turned out to be true ... you used that fact in your alleged "proof" of that fact. So the proof is circular.

Define a cover C on X: C = { {n} | n is an integer } ∪ { {TREE(3)} }

Are you implying that TREE(3) is not an integer?

More insanity.

Of course you can take refuge by noting that set union is not necessarily disjoint. But why else write it that way?

Curious about one thing. Do they teach this watered down version of sheaves to CS majors these days? Interesting if true.

ps -- Are you implying that TREE(3) is the ONLY integer with indeterminate parity? All the others have determinate parity? Is that your claim in this silly "proof" which doesn't prove anything?

[Skepdick]

For TREE(3), the section over {TREE(3)} (extend this section as needed) maps TREE(3) to 'undetermined' preserving our epistemic intuitions.

This is what they call assuming that which is to be proved.

Eh? I am not assuming that which is to be proved. The parity of TREE(3) is to be proved.

Have you proven it to be Odd or Even yet? I haven't. So how would you like me to represent or express the value of MOD2(TREE(3)) ?

You seem to believe TREE(3) has no determinate parity. That's wrong, if not flat out insane.

You seem to believe that TREE(3) has a determinate parity like 4 has determinate parity. That's wrong. If not flat out insane.

I can model/represent the value of MOD2(4) as 0.
If you don't want me to model/represent the value of MOD2(TREE(3)) as 'undetermined' then how would you like me to model/represent it?

But even if it turned out to be true ... you used that fact in your alleged "proof" of that fact. So the proof is circular.

What is it that has to "turn out to be true"? The fact that the value of the expression "MOD2(TREE(3))" is undetermined?

It's obviously true. Nobody has any clue how to determine its value!

Are you implying that TREE(3) is not an integer?

More insanity.

Of course you can take refuge by noting that set union is not necessarily disjoint. But why else write it that way?

Oh no! A frivolous step! Are you running out of things to nitpick?

Curious about one thing. Do they teach this watered down version of sheaves to CS majors these days? Interesting if true.

They don't teach it to CS majors. Distributed Systems Engineers figured out that the line between theory and practice is blurred because... "In mathematics, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them."

That sounds like fancy mathematical jargon for a control plane between interacting interacting systems.

ps -- Are you implying that TREE(3) is the ONLY integer with indeterminate parity? All the others have determinate parity? Is that your claim in this silly "proof" which doesn't prove anything?

Eh? I am painting in broad strokes here. TREE(3) is but one of many integers for which you can't determine which of these parallel computations wil halt:

A: (MOD2(X) == 0) halts ONLY if True
B (MOD2(X) == 1) halts ONLY if True

You want me to prove that I have neither a halting proof for A nor a halting proof for B ?!?!? The fuck?

The above holds for any integer-encoding for which I can't obtain a full expansion. You know - parity is nothing other than the least significant bit.

[Magnus Anderson]

Have you proven it to be Odd or Even yet? I haven't. So how would you like me to represent or express the value of MOD2(TREE(3)) ?

It has been proven more than one here in this thread that TREE(3) is either even or odd. And that's all that has to be proven. Which one of the two it is is irrelevant -- merely a distraction of yours. It's called burdening the other side with unnecessary tasks.

[Magnus Anderson]

It's either 0, or 1, or 2; or 3; or 4; or 5; or 6; or 7; or 8; or 9; or 0.

And that means it's either even or odd.

Thank you very much.

[Skepdick]

Have you proven it to be Odd or Even yet? I haven't. So how would you like me to represent or express the value of MOD2(TREE(3)) ?

It has been proven more than one here in this thread that TREE(3) is either even or odd.

A proof entails telling me which disjunct holds.
Parroting an axiom is not a proof.

And that's all that has to be proven. Which one of the two it is is irrelevant -- merely a distraction of yours. It's called burdening the other side with unnecessary tasks.

OK! I've proven that everything in the universe is either an Elephant or it isn't.

[Skepdick]

It's either 0, or 1, or 2; or 3; or 4; or 5; or 6; or 7; or 8; or 9; or 0.

And that means it's either even or odd.

Thank you very much.

It also means itss either an elephant or not.
It also means it's either a toenail or not.
It also means it's a cup of tea or not.
It also means it's a dildo or not.
.....

It also means P=NP is either true or false.

Tautological tautology is vacuous.

[Magnus Anderson]

It also means itss either an elephant or not.
It also means it's either a toenail or not.
It also means it's a cup of tea or not.
It also means it's a dildo or not.
.....

Correct. The only problem is that you do not seem to have a point.

You're arguing that TREE(3) is neither even nor odd. I am telling you that it has been proven here in this thread that it's either even or odd, i.e. the opposite of what you're claiming.

You not knowing whether TREE(3) is even or odd is not a proof that it's neither even nor odd. That's called "argument from ignorance" and it's a logical fallacy.

That's where your career in nonsense ends.

[Magnus Anderson]

I gave you a proper proof.

1. There is NO proof for it being odd.
2. There is NO proof for it being even.
3. There is NO reason to believe it's either odd OR even because NEITHER one is proven true.

The statement "There is NO reason to believe it's either even or odd" is not the same as "It's neither even nor odd".

Learn the difference first.

Next, there is very much a reason to believe it's one of the two ( even or odd. )

It's been proven multiple times here in this thread, you ignoring it and pretending like it never happened does not change that fact.

An integer is said to be even if the last digit of its decimal representation is 0, 2, 4, 6 or 8.

An integer is said to be odd if it's not even.

End of story.

By definition, it's one of the two.

[Magnus Anderson]

A proof entails telling me which disjunct holds.
Parroting an axiom is not a proof.

Again, if you operate with your imaginary, made up, fantasy version of logic, nothing can be resolved until you learn how to properly think.

You rejecting perfectly valid proofs does not mean they are invalid.

It merely means you have yet to learn how to think.

OK! I've proven that everything in the universe is either an Elephant or it isn't.

And? Are you a teletubby?

[Skepdick]

It also means itss either an elephant or not.
It also means it's either a toenail or not.
It also means it's a cup of tea or not.
It also means it's a dildo or not.
.....

Correct. The only problem is that you do not seem to have a point.

The point (which has gone over your head 10 times now)

Is that OR is a logical disjunction.
A disjunction in which you don't know which disjunct holds is of no value.

You're arguing that TREE(3) is neither even nor odd. I am telling you that it has been proven here in this thread that it's either even or odd

Which was proven? Orr; or Even?

i.e. the opposite of what you're claiming.

You are using a Classical frame to misinterpret a Constructive claim.

You not knowing whether TREE(3) is even or odd is not a proof that it's neither even nor odd.

You are using a Classical frame to misinterpret a Constructive claim.

That's called "argument from ignorance" and it's a logical fallacy.

You are using a Classical frame to misinterpret a Constructive claim.

It is not a fallacy to assert that you have proven neither TREE(3)'s oddness nor its evenness.

It's not an argument FROM ignorance.
It's an argument TO ignorance.

You are ignorant of TREE(3)'s parity-value.

That's where your career in nonsense ends.

That's no career I want to take away from you.

[Skepdick]

I gave you a proper proof.

1. There is NO proof for it being odd.
2. There is NO proof for it being even.
3. There is NO reason to believe it's either odd OR even because NEITHER one is proven true.

The statement "There is NO reason to believe it's either even or odd" is not the same as "It's neither even nor odd".

Learn the difference first.

There is no difference between these two statements in a Constructive frame.

Its oddness is unproven.
Its evenness is unproven.

There's no reason to believe it's either odd or even.

Next, there is very much a reason to believe it's one of the two ( even or odd. )

Which one and how much reason?

By definition, it's one of the two.

Which one?

[Skepdick]

Again, if you operate with your imaginary, made up, fantasy version of logic, nothing can be resolved until you learn how to properly think.

And if you operate with your imaginary, made up, fantasy version of logic something can be resolved before you do any thinking.

You rejecting perfectly valid proofs does not mean they are invalid.

There aren't any proofs to reject.

0 proofs for Oddness + 0 proofs for evenness = 0 proofs in total.

It merely means you have yet to learn how to think.

And? Are you a teletubby?

Everything is either a teletubby or it isn't!

[Magnus Anderson]

The point (which has gone over your head 10 times now)

Is that OR is a logical disjunction.
A disjunction in which you don't know which disjunct holds is of no value.

It very much is. It tells us that every integer is either even or odd which is the opposite of your claim that TREE(3) is something else ( neither even nor odd. )

You also keep ignoring that "Neither even nor odd" is self-contradictory since it translates to "Odd and not odd" given that "not even" means "odd".

You are, once again, refusing to actually think.

Which was proven? Orr; or Even?

It has been proven more than once here in this thread that it is one of the two -- even or odd -- rather than something else.

Quite contrary to your constructive mathematics bullshit, there is no need for anyone to prove whether TREE(3) is even or odd. Proving that it MUST be one of the two is ENOUGH.

Making things artificially and unnecessarily difficult is not a sign of intelligence.

It is not a fallacy to assert that you have proven neither TREE(3)'s oddness nor its evenness.

You're claiming that you've proven that TREE(3) is neither even nor odd.

Learn the difference between "I don't know if it's even or odd" and "I know it's neither even nor odd".

It's not an argument FROM ignorance.

You wish.