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I've been reading a book on Set Theory (Charles C. Pinter), and it says,

...set theory is recognized to be the cornerstone of the "new" mathematics... [emph. added]

and that

...we can still form all the sets essential for mathematics... [emph. added]

Background

The book also states that Set Theory can be used to form the framework for mathematics (I am regrettably unable to quote Pinter on that). For example, if $A$ and $B$ are two disjoint sets, and that $a=\# A$ and $b=\# B$ denote that $a$ and $b$ are the cardinal numbers representing $A$ and $B$ respectively, then: $$a+b=\#(A\cup B),$$

and that $$a\cdot b=\#(A\times B).$$

Exponentiation can be produced by:

Let $a$ and $b$ be cardinal numbers, and let $I$ be a set such that $b=\#I$. If $a=a_i$,$\forall i\in I$, then:

$$a^b=\bigotimes_{i\in I}a_i.$$

where

$$\bigotimes_{i\in I}a_i=\#\left(\prod_{i\in I}A_i\right),$$

when $A_i$ is a family of sets, $a_i=\# A_i, \forall i\in I$.

Question

This, oddly enough, is where Pinter decided to stop. He did not say how to produce division, or even the subtraction of cardinal numbers. I guess the subtraction of a cardinal would be (assuming that $0=\varnothing$, $1=\{\varnothing\}$, $2=\{\varnothing,\{\varnothing\}\}$... or that $0=\varnothing$, $1=\{0\}$, $2=\{0,1\}$...), and if $a=\#A$ and $b=\#B$: $$a-b=A\setminus B,$$ but this, to my knowledge, only works when $a\geq b$, for one cannot have a cardinal (let's say $x$) where $x<0$ (because that means the set is a proper subset of $\varnothing$, which is impossible).

How can I use Set Theory to represent functions like subtraction and division?

Xander Henderson
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Conor O'Brien
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    You can formally construct the integers with an equivalence relationship atop the natural numbers, then construct the rationals using the integers you've just constructed. – Alex Nelson Dec 08 '14 at 02:32
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    And, indeed, it is impossible to define subtraction or division on cardinal numbers in any meaningful way once you get to infinite sets (which is the point at which the true power of set theory manifests itself). – Peter Košinár Dec 08 '14 at 02:53
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    I guess that those questions are related:

    • http://math.stackexchange.com/questions/146844/how-to-divide-aleph-numbers/

    • http://math.stackexchange.com/questions/140930/cardinal-number-subtraction/140947#140947

    • http://math.stackexchange.com/questions/920468/division-of-cardinals

     – MphLee Dec 08 '14 at 12:48
  • Conor, have you looked at the links that @MphLee posted? If you're asking about divisions of cardinal numbers, and it seems that this is what you're asking about, then the topic was covered already. For infinite sets this notion becomes either too trivial or not well-defined; for finite sets, well we can formally define the non-negative rational numbers and now say that the cardinals are a subset of that; but what good comes out of this? This won't correspond to any natural notion of cardinality of a set anyway. – Asaf Karagila Dec 08 '14 at 21:22
  • @AsafKaragila I never did get Aleph Numbers... – Conor O'Brien Dec 08 '14 at 21:27
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    Those are just infinite cardinals. Assuming the axiom of choice these are all the infinite cardinals, too. But without the axiom of choice, everything about the structure of cardinals (including much of their arithmetic, except the basic definitions) can break down. – Asaf Karagila Dec 08 '14 at 21:33
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    You actually want $a+b=#(A\times{0}+B\times{1})$ for sets $0\ne1$. – J.G. Jan 26 '21 at 15:16
  • You could use quotient sets, but I am not sure it would make much sense (basically with axiom of choice, you could surject A into B in a right way, and use the equivalence relation created by this surjection to quotient A. But I am not sure it makes a lot of sense…) – tbrugere Jan 26 '21 at 16:23
  • as a side comment, you actually can write a big times symbol in mathjax. try for instance this code {\large{\times}}_{i\in I}a_i, which gives ${\large{\times}}_{i\in I}a_i$. use \Large instead of \large if you would like it even larger. – Atticus Stonestrom Jan 26 '21 at 16:26
  • I have cleaned up your question a little, and replaced much of the html with Markdown. Note that mathematicians are a little more lax about precisely quoting authors than other researchers---it is generally sufficient, when recalling definitions, to cite a work and then introduce the definitions. Changes in notation are nothing to be bothered about, and taking care to precisely quote the source (with ellipses and everything) is not necessary. – Xander Henderson Jan 29 '21 at 00:42
  • Thanks for that note @XanderHenderson, I've come far and learned much in 6 years :) – Conor O'Brien Jan 29 '21 at 03:45

1 Answers1

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But how do you define subtraction or division anyway?

$a=c-b$ if and only if $a+b=c$

The same

$a=c/b$ if and only if $a \cdot b=c$

That is the most general definition possible.

Subtraction and division might not have a solution as they do not, for example, if you restrict to positive integers and integers respectively.

So you can define subtraction and division, yet the fact that you cannot perform that operation for all possible sets out there is equally valid.

Alex Peter
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