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Comment by dr_dshiv | Hacker News Reader

Can I ask how sets come before numbers?

And, does the concept of one/whole exist before sets?

Set theory constructs the natural/ordinal numbers from it's axioms, but set theory is a man-made formal language that didn't exist until the 19th century, so talking about what comes/before after is kind of moot.

https://en.wikipedia.org/wiki/Ordinal_number

It not about historical order, it's about the order in which you need things to build upon mathematically.

You don't need set theory for numbers, enumeration or arithmetic.

You don't need it, but sets are more primitive than numbers.

For example, the category of finite sets without an NNO [1] is simpler or more foundational than the set of natural numbers. At the same time, this category is actually a category of numbers.

My point is that numbers are complicated by nature, but they are more intuitive for humans (mostly I think) than sets are. Sets are simpler or more basic, but often less intuitive for humans.

[1] https://en.wikipedia.org/wiki/Natural_numbers_object

I think you can argue that the idea of finite sets is prior to numbers, though the machinery of set theory is obviously much newer. When you count a field of sheep, you need some idea of the set if sheep you are counting.

It was a kind of Peano arithmetic with pebbles: you kept a sack or pile of pebbles and moved them to another sack or pile one-by-one as your sheep went by one-by-one (Ye Olde Pigeon Hole Principle) and that way you knew you had all of your sheep. Two major innovations after that were the idea that you could e.g. add two piles of pebbles to "calculate" the size of combining two flocks of sheep into a larger flock, and so on, and the idea of naming the pebbles (perhaps from a "counting" song) and then letting the names themselves substitute for the actual pebbles.

So, yes, idea of finite sets is prior to numbers, but numbers were formalized before the idea of sets was formalized, IMO, FWIW.

One begins with nothing. Specifically, the set containing nothing: {}. From there, we can describe a sequence of sets, each containing the previous one: {{}}, {{{}}}, {{{{}}}}, etc. These sets can then be associated with natural numbers, though arithmetic in this context becomes somewhat perverse.

1 := {}

2 := {{}}

...

n+1 := {n}

This is the simplest set-based description of the natural numbers that I know, and it's very unwieldy -- one might like to construct negative numbers and zero; describe algorihms to perform arithmetic, etc. But it's all quite manageable and a satisfying exercise if one takes the time.

I believe a more standard approach is to say

0 := {}

1 := {{}} (= {0})

2 := {{}, {{}}} (= {0, 1})

3 := {{}, {{}}, {{}, {{}}}} (= {0, 1, 2})

n+1 := {0, 1, 2, ... n}

This has the advantage that the numbers are equal to the cardinality of the sets that define them. In the construction you gave (which I agree is conceptually easier), the cardinality of 1 is 0, and then the cardinality of every greater integer is 1.

A cool feature of this version is

x < y := x ∈ y

One begins with nothing... How nice! It seems quite ridiculous to ask which came first.

Cardinal numbers are sets. And after that comes ordinals, which are enumerated numbers.

[1] https://en.wikipedia.org/wiki/Cardinal_number [2] https://en.wikipedia.org/wiki/Ordinal_number

Von Neumann Construction https://en.wikipedia.org/wiki/Von_Neumann_universe

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