Talk:Equinumerosity

Notation

Notation question: the author of this page uses the following notation for equinumerous sets: $A\approx B.\,$ Is this more correct than: $A\sim B.\,$ which is what is used in Apostal, "Mathematical Analysis". Lemasterda 16:49, 14 October 2007 (UTC)

I have seen the former used more often than the latter for this meaning. JRSpriggs 17:02, 14 October 2007 (UTC)

Issues With The Term Equinumerosity

I think the term Equinumerosity is too strong of a word to describe two sets that have a one-to-one bijection among themselves. In the youtube video "can something be twice as big as itelf" http://www.youtube.com/watch?v=URAvLTbBcHY&list=UUw74HOUKLmAmw-IY6YO7YIQ&index=25 I point out that a set can have a many-to-one bijection with another set, while still having a one-to-one bijection with that same set. So rather than describe these sets as equinumerous, I suggest one is still bigger than the other because it has a bigger measure. I suggest that because it has a bigger measure, it has a higher number of elements. Therefore, the sets are not equinumerous. I suggest that sets with uncountably infinite elements are MATCHLESSLY bigger than sets with countably infinite elements; while certain sets with countably infinite elements can be bigger than other sets with countably infinite elements, just not MATCHLESSLY bigger.

To demonstrate this equinumerosity issue, I created the dead celebrities paradox, where I compare the deaths of two celebrities. Michael Jackson and Elvis are both dead celebrities. Elvis died before Michael Jackson. The question is, which celebrity will be dead the most number of seconds? Both will be dead forever, but which forever is longer, the forever that started earlier with Elvis's death, or the forever that started with Michael Jackson's death? Traditional cardinality theory suggests that they would both be dead the exact same number of seconds because of the one-to-one bijection, but I'd rather suggest that our intuitive ideas are correct and that Elvis will always be dead for more seconds than Michael Jackson will be. The sets of seconds are not equinumerous.

Theboombody (talk) 14:44, 21 April 2013 (UTC)

"many-to-one bijection" is a contradiction in terms. Therefore, I cannot answer your question because it makes no sense. JRSpriggs (talk) 15:37, 21 April 2013 (UTC)

I define a many-to-one bijection as a set that is many-to-one and onto (surjective) rather than the traditional definition of bijection which is one-to-one (injective) and onto (surjective). I could invent a new term to prevent ambiguity, but I prefer using an altered version of the traditional term. The question should now make more sense now that a many-to-one bijection has been defined.

Theboombody (talk) 02:38, 4 May 2013 (UTC)

It appears that your "many-to-one bijection" is a surjection which is not an injection. Please use standard terminology, if you want to have a rational discussion. JRSpriggs (talk) 03:02, 4 May 2013 (UTC)

I never claimed many-to-one was an injection. I know it's not. Perhaps I can show you an example of what I mean by many-to-one. The sets (1,2,3) and (4,5,6) are equivalent in their size if their size is measured by their number of elements because a bijection exists between them. A one-to-one and onto relationship can be established between them. (1,2,3,4,5,6,7,8,9) and (10,11,12) are not equivalent in size. No bijection can be made. Instead, a three-to-one and onto relationship can be established between them. So one of the sets is exactly three times bigger than the other, no more, no less. That's what I mean by many-to-one. Always surjective, never injective. But it clearly shows that one of the sets is larger than the other by a certain whole number multiple. This has been my primary focus in challenging the established conventions of equinumerosity among certain infinite sets.

So whether or not my language is valid or respected is not as important as the idea behind them. Many-to-one, surjective relationships among sets clearly demonstrate their non-equivalence.

Theboombody (talk) 03:23, 4 May 2013 (UTC)

It is essential to use the correct language, because otherwise we will be arguing at cross purposes. You will be trying to refute things which I did not say, and I will be trying to refute things which you did not mean.

To the substance: The problem with your ideas is that you are trying to apply to infinite sets ideas which only apply to finite sets. The differences between finite sets and infinite sets are quite extreme, see Finite set#Necessary and sufficient conditions for finiteness for some of the many ways they are different (just imagine the opposite of those conditions to see what infinite sets are like). JRSpriggs (talk) 03:50, 4 May 2013 (UTC)

I agree that my idea that I am applying to infinite sets really only works with finite sets and I have no justification for extrapolating these ideas to infinite sets, but the same can be said of the definition of a bijection itself. The only reason (as far as I know) that a bijection shows equivalency of sets between certain infinite sets is because bijections work with finite sets and their application has been extrapolated to sets that aren't finite. That's it. Cantor was trying to find a way to measure infinite sets, ran into road blocks naturally, and invented the definition of a bijection only because a bijection is a good way to show equivalence between finite sets. So he took what worked with finite sets and extrapolated it to the infinite. Nothing more than that. I think if he can do extrapolation like that with his definition of injective, then surely someone can do the same extrapolation with a many-to-one idea. Why not? How can you extrapolate one-to-one and then forbid extrapolation of many-to-one? I don't think you can just pick and choose what to extrapolate from finite to infinite. Either extrapolate everything or extrapolate nothing.

Theboombody (talk) 14:40, 4 May 2013 (UTC)

The determination of which properties extend to infinite sets and which do not is not arbitrary. I am still trying to figure out what is the best way to explain it to you. JRSpriggs (talk) 20:07, 5 May 2013 (UTC)