/sci/ - Science & Math » Thread #13307703

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Can two objects be described in exactly the same way, yet be different?

Anonymous Mon 21 Jun 23:02:34 2021 No.13307703 View Reply Original Report

Quoted By: >>13307754 >>13307755 >>13308787 >>13308800

Every mathematical object (construction) is defined as something that fulfills a set of conditions so my guess would be that there would have to be "hidden" conditions that make those two objects different from each other.

Anonymous

Anonymous Mon 21 Jun 2021 23:21:44 No.13307754 Report

Quoted By: >>13307797

>>13307703
Yes, provided the description doesn't define the object uniquely. For example you could say let n be a natural number. That was a description of n, but both 2 and 3 fulfill that description, yet 2!=3.

Anonymous

Anonymous Mon 21 Jun 2021 23:21:46 No.13307755 Report

Quoted By: >>13307797

>>13307703
you just mentioned one of the "hidden" conditions

Anonymous

Anonymous Mon 21 Jun 2021 23:37:35 No.13307797 Report

Quoted By: >>13308744

>>13307755
That there exist hidden conditions?
>>13307754
I was talking about a description that defines an object uniquely, yes

Anonymous

Anonymous Tue 22 Jun 2021 02:54:35 No.13308429 Report

Quoted By:

bump

Anonymous

Anonymous Tue 22 Jun 2021 04:25:12 No.13308691 Report

Quoted By:

No. Identity of indiscernibles, you fool

Anonymous

Anonymous Tue 22 Jun 2021 04:30:54 No.13308722 Report

Quoted By:

34.629999999999999 = 34.63

Anonymous

Anonymous Tue 22 Jun 2021 04:36:28 No.13308744 Report

Quoted By:

>>13307797
It depends what you mean by unique definition. This could either mean that the definition only describes one object or that the object only has one definition. If the object only has one definition then that definition could conceivably define multiple objects, although it's hard to think of an example.

Anonymous

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Anonymous Tue 22 Jun 2021 04:37:50 No.13308755 Report

Quoted By:

That depends on where they are described. Take the spaces $X:= \bigvee_{n\ge1} S^{2n}$ and $\mathbb{C}P^\infty$ for example. Looking at the cohomology groups, we have isomorphisms $H^n(X; \mathbb{Z}) \cong H^n(\mathbb{C}P^\infty; \mathbb{Z})$ , so you can't distinguish between the two of those if you were describing them using only cohomology as a graded abelian group. If you, however, take cup products into account, you have a "hidden condition" showing they aren't equal. Fill in the details as a nice morning exercise.

Anonymous

Anonymous Tue 22 Jun 2021 04:47:50 No.13308787 Report

Quoted By:

>>13307703
Just place a well ordering on the set of all sets desu

Anonymous

Anonymous Tue 22 Jun 2021 04:51:42 No.13308800 Report

Quoted By:

>>13307703
Yes, in a certain sense. For example, you could have two mathematical structures that are isomorphic as groups, but not isomorphic as rings.

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