# Is the following simplification of the von Neuman natural number construction legit?

I’m giving a really brief talk about constructing math from first principals (basically I just want to give people of feel for it) and I wanted to mention how you can derive the natural numbers starting from the empty set.

Von Neumans construction was

0=Null
1={Null}
2={{Null},{{Null}}}
3={{Null},{{Null}},{{{Null},{{Null}}}}
etc.

In order to go over this quickly wanted to simplify this to

0=Null
1={Null}
2={{Null}}
3={{{Null}}}
etc.

I don’t think there is anything wrong with this, but I’m not an expert in this area and didn’t want to say something mathematically incorrect.

Yes, this works fine, and was historically considered; the advantage of the von Neumann construction is that it extends just as well to the transfinite ordinals (thus, omega, the first transfinite ordinal after all the naturals, can be taken to be the infinite set {Null, {Null}, {Null, {Null}}, …}, continuing the same pattern (that each ordinal is the set of all previous ordinals)).

[Personally, I’m not particularly keen on encoding everything as sets of sets of sets of sets…, but it seems to make some people happy.]

That is, the drawback of the construction taking each finite ordinal to be the singleton containing its predecessor (with predecessor-less zero as a special case) is that it does not extend cleanly to the transfinite ordinals, where many further ordinals also lack predecessors (the so-called “limit ordinals”, such as the aforementioned omega).

What is the second construction called?

The Xzibit Construction.

It’s the Zermelo construction of the natural numbers. But people don’t talk about it very much.

Another advantage of the von Neumann construction is that n has n elements, not just one.

My late colleague claimed that 2 is the cardinality of the set consisting of a pair of platinum balls kept in a vault in Sevres, France.

Are you sure you don’t have too many braces? I thought 2, 3 should be

2={Null, {Null}}
3={Null, {Null}, {Null,{Null}}

or more simply
K+1 = K ∪ {K}

Yes, septimus is correct; in the von Neumann construction, each ordinal is the set of all previous ordinals.

Yo dog I herd u liek sex so I put a set in your set so you can have lot of sets.

OK thanks all,

I wanted to use this construction so I could claim that I was deriving all of mathematics starting with “nothing”, (i.e. the empty set).
For this talk I have no intention of discussing the transfinite ordinals, the talk about infinity will be next year.

The natural numbers are hardly all of mathematics. And we know you can’t derive all (and only) mathematics. The claim you made in the OP (“you can derive the natural numbers starting from the empty set”) was accurate.

In one standard model of set theory, take any set, then take any element of that set, which is a set, since everything is, then take any element of that set, continue as long as you can. In a finite number of such steps, you must eventually reach a set with no elements, that is the empty set.