Timeline for Structural definition of "product" in set theory
Current License: CC BY-SA 3.0
21 events
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| Oct 22, 2013 at 15:04 | history | edited | Hans-Peter Stricker | CC BY-SA 3.0 |
deleted 13 characters in body
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| Oct 10, 2011 at 12:55 | answer | added | Buschi Sergio | timeline score: 0 | |
| Oct 10, 2011 at 9:23 | comment | added | David Roberts♦ | "I don't think that ZFC style set theory on its own is really designed to be a handy framework for actually doing maths." +1 :) | |
| Oct 10, 2011 at 8:16 | answer | added | Martin Brandenburg | timeline score: 5 | |
| Oct 9, 2011 at 21:10 | comment | added | George Lowther | @Hans Stricker: I see that you are trying to define the concept of a product without refering to its individual components, which my previous comment missed. Not sure why though. | |
| Oct 9, 2011 at 21:07 | comment | added | George Lowther | @Hans Stricker: I see that you are trying to define the concept of an ordered pair without defining the components of the pair itself, which my previous comment missed. Not sure why though. | |
| Oct 9, 2011 at 20:58 | comment | added | Andreas Blass |
To amplify a-fortiori's comment: The second clause in your definition says, when the notation is understood as explained right after the definition, that for each $z$ there is just one $x$ with just one $y$ such that $z$ agrees with $x$ in the first component and with $y$ in the second. In fact, there are as many such $x$'s (resp. $y$'s) as the cardinality of the quotient $S/\sim_1$ (resp. $S/\sim_2$). The first component of $x$ must agree with that of $z$ but the second is arbitrary; similarly for $y$.
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| Oct 9, 2011 at 19:30 | comment | added | George Lowther | @Hans Stricker: Right, so I partly agree with you there. Except, rather than adding a pair of relations, I think adding a 'Pair' keyword is neater. This is a 2-ary function symbol satisfying the axiom $$\forall x_1,x_2,y_1,y_2\;\left(({\rm Pair}(x_1,x_2)={\rm Pair}(y_1,y_2))\rightarrow(x_1=y_1 \wedge x_2=y_2)\right).$$ Or, add another pair $\pi_1,\pi_2$ of 1-ary functions satisfying $$\forall x,y\;\left(\pi_1{\rm Pair}(x,y)=x\wedge\pi_2{\rm Pair}(x,y)=y\right).$$ In any case, I don't think that ZFC style set theory on its own is really designed to be a handy framework for actually doing maths. | |
| Oct 9, 2011 at 17:17 | comment | added | Hans-Peter Stricker | @George: "what matters is the relations between the objects" - that's what I tried to capture with my definition: ordered pairs are atomic objects, kept together by some relations. | |
| Oct 9, 2011 at 15:35 | comment | added | Will Sawin | The category theory also makes it clear why there can be no acceptable definition. Because a product is an object with structure (in the form of two functions), and there is no intrinsic definition for structure. | |
| Oct 9, 2011 at 13:08 | comment | added | KConrad | The product of sets, together with its projection maps to the factors, could be characterized by a suitable universal mapping property. Then that construction is unique up to suitable isomorphism as all objects satisfying universal properties are. This can let you relax a little about the fact that there may be more than one way to make the construction of a product of sets (and its projections to the factors). | |
| Oct 9, 2011 at 9:59 | comment | added | George Lowther | And I'd think of the product $X\times Y$ in terms of the projections $X\times Y\to X$, $X\times Y\to Y$. Of course, functions/homomorphisms are not part of the definition of ZFC, so have to be constructed --- usually after the construction of pairs and products. | |
| Oct 9, 2011 at 9:52 | comment | added | George Lowther | I would regard the `definition' of ordered pair in ZFC (or similarly) merely as a proof of existence of ordered pairs/products. Just as there are many constructions of the natural numbers, integers, rationals, real and complex numbers. Really, what matters is the relations between the objects. | |
| Oct 9, 2011 at 9:32 | comment | added | Hans-Peter Stricker | I just wanted to say that there are many definitions of "ordered pair" and none of them is distinguished. | |
| Oct 9, 2011 at 9:26 | comment | added | Not Mike | I understand that I might be a bit sheltered, but: at what point did the definition of ordered pair become 'notoriously arbitrary'? | |
| Oct 9, 2011 at 8:15 | comment | added | user2035 | The second condition is not satisfied by usual products in general. | |
| Oct 9, 2011 at 8:09 | comment | added | Hans-Peter Stricker | Is the empty set case the only problematic one (in this context)? | |
| Oct 9, 2011 at 8:08 | comment | added | Hans-Peter Stricker | @Andrej: Yes, of course I did mean $(\forall x \in S)$ etc. I hoped this would be clear from the context. | |
| Oct 9, 2011 at 8:01 | comment | added | Andrej Bauer | The definition of a product should involve three sets, not just one. If I tell you that $P$ is a product of two sets, you cannot always recover from $P$ the two sets (think of the case when one of the sets is empty). Your definition is unlikely going to work when one of the components is an empty set. How would we get $\emptyset \times \mathbb{N}$, for example? If you take $S = \emptyset$, as you presumably should, then you cannot recover $\mathbb{N}$. | |
| Oct 9, 2011 at 7:58 | comment | added | Andrej Bauer | In your definition of a product, did you mean $\forall x \in S \forall y \in S \exists! z \in S \ldots$ instead of the unbounded quantification $\forall x \forall y \exists! z \ldots$? | |
| Oct 9, 2011 at 7:37 | history | asked | Hans-Peter Stricker | CC BY-SA 3.0 |