Timeline for Mathematical objects existing as different instances
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Dec 21, 2021 at 11:15 | comment | added | Rushi | You may find this humorous account of the infinite "copy-ability" of math objects interesting microsoft.com/en-us/research/publication/… | |
| Dec 21, 2021 at 1:35 | comment | added | user4894 | @DavidGudeman If you have two distinct sets like N and {0} you can surely form their Cartesian product N × {0} (not their "cross" product, which is something else entirely that doesn't apply in this context). But if you have a single set like N, by what principle of set theory do we obtain "another copy" of N with which to form the Cartesian product N × N? As I noted, nobody ever thinks twice about this ... but if you DO happen to think twice, it's a good question. What are two "copies" of the same set in the context of formal set theory? I gave the answer. | |
| Dec 20, 2021 at 17:31 | comment | added | David Gudeman | I'm confused by this answer. You seem to be using cross products to justify cross products. I'll note that normally, in set theory, the ordered pair (x,y) is represented with the set {x, {x,y}}. Is that perhaps what you had in mind? | |
| Dec 19, 2021 at 13:56 | comment | added | J D | Notice this is just a special case of all addition such that a + b := (...((x))) : x=(...((0))). :D | |
| Dec 19, 2021 at 13:54 | comment | added | J D | You'll note that the ordered pair (2,2) derived from the Cartesian product implies (2,2*) since (a,b,c,...) by definition is a mapping of the S of elements onto N and aligns nicely with the extension of PA where a + a* implies that a* is defined in terms of a, since a then functions as a unary operator. | |
| Dec 19, 2021 at 13:49 | comment | added | J D | @user1007028 2 + 2 is defined as an extension of PA. Roughly, 2 is a defined as a recursive function of successors of 0 so that 2 := ((0)) and addition is the the substitution of the adder for 0 in the addend such that 2 + 2 := ((x)) : x=((0)) so that 2 + 2 = ((((0)))) which from PA is 4 by definition. | |
| Dec 19, 2021 at 12:38 | comment | added | J D | Unless I'm wrong, you can ground your example in two logical axioms of ZFC, on the one hand the axiom of infinity guarantees us an infinite number of sets, and on the other hand, the axiom of extensionality assures us if there's a difference in definition, then we simply characterize the "copies" with some other criterion. For instance, R mapped onto R is really r in R and r* in R* such that there exists (r,r*). | |
| Dec 18, 2021 at 23:06 | comment | added | Confused | thanks for the explanation anyway | |
| Dec 18, 2021 at 23:06 | comment | added | user4894 | I might be overeducated rather than educated on this particular issue. I'm honestly not 100% certain that one is required to formalize a binary operation the way I did in order to say that 2 + 2 = 4. But I am 90% certain. Otherwise what is a binary operation? It must input an element of the Cartesian product. So that's the basis of my belief. But I could be wrong, and I haven't much to add. | |
| Dec 18, 2021 at 23:03 | comment | added | Confused | I would have generally interpreted something like 2+2 to be an operation on the exact same thing by examining it's properties as a concept one concept only no copies, but like I said this becomes pointless when looking at it the way you are, and I'm not sure if I could take what you've said and apply this way of thinking to it (you are probably much more educated on this than me), but my main issue is literally with the semantics of an expression 1+2, do we consider this just me writing a particular copy of 1 and a particular copy of two and getting a particular copy of 3? | |
| Dec 18, 2021 at 23:00 | comment | added | user4894 | (more) But I can't be certain that this is absolutely required. Maybe I'm going overboard. I'm actually not entirely sure.But if you need to do this trick to form the Cartesian product of a set with itself, it must be the case that you need to do this to define a binary operation on a set. | |
| Dec 18, 2021 at 22:59 | comment | added | user4894 | That's a good question. If you have a set such as the set of integers, addition is a binary operation on the set. That is, the operation inputs a pair of integers and outputs their sum. In order to formalize this idea you would have to say that addition is a function from the Cartesian product of the integers with themselves to the integers. To form the Cartesian product you need the trick I showed. But I'm not 100% sure this can't be done more simply. A harder problem is 2 + 2. Where do you get "two twos" to add? Again, it must be that we are making a secret copy using the same trick. | |
| Dec 18, 2021 at 22:55 | comment | added | Confused | This is very interesting, how would we consider an expression of two numbers then how would we talk about, an expression of the combination of two numbers such as 1+2, I would generally think of having one set of individual objects which is then applied to each member co-ordinate (x,y), but this has definitely made me curious as a way of looking at this, because before encountering multi-sets I would have said a number is simply one object that is applied to a variable, position in an expression etc to perform calculations on single objects. | |
| Dec 18, 2021 at 22:03 | history | edited | user4894 | CC BY-SA 4.0 |
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| Dec 18, 2021 at 21:58 | history | edited | user4894 | CC BY-SA 4.0 |
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| Dec 18, 2021 at 21:52 | history | answered | user4894 | CC BY-SA 4.0 |