Badiou's Being and Event and the Mathematics of Set Theory by Baki Burhanuddin;

Author:Baki, Burhanuddin;
Language: eng
Format: epub
ISBN: 9781472578716
Publisher: Bloomsbury UK
Published: 2019-11-22T16:00:00+00:00

An ordinal is the set of the ordinals preceding it

Now there is another property of the ordinals that we have postponed mentioning. Except for 0, every ordinal number is defined as equivalent to the set of ordinals preceding it. In other words, the ordinals are exactly equal to the initial segments we mentioned. The ordinal 6, for example, is the set of ordinals from 0 to 5:

6 = {0,1,2,3,4,5}.

So, in general, every natural number n, when conceived as an ordinal, is precisely the set of all natural numbers preceding n. So n = {0,1,2, …, n} for every n ∈ ℕ, and likewise for the infinite ordinals λ = {x: x < λ} for every ordinal λ except for 0. Thus, every ordinal is identical to the sequence of ordinals preceding it. Since every order type can be matched to an initial segment of the ordinals and every initial segment is, in fact, an ordinal itself, we have therefore provided a match between the order types and the ordinals.

Now the principle behind set theory is that every mathematical object is a set. As given previously, an ordinal is simply the set of all ordinals preceding it:

1 = {0}

2 = {0,1}

3 = {0,1,2}

4 = {0,1,2,3}

⋮

ω = {0,1,2,3, …}

ω + 1 = {0,1,2,3, … ω}

⋮

Since every ordinal refers back to all the ordinals before it, every ordinal ultimately comes back to the first ordinal, namely 0. Hence, each ordinal is built from zero. The set-theoretic construction is completed with the decision to equate the ordinal 0 with the empty set ∅ itself. This makes sense since zero and the empty set are simply different proper names for the void itself.

0 = ∅

1 = {∅}

2 = {∅, {∅}}

3 = {∅, {∅}, {∅, {∅}}}

4 = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}

⋮

ω = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, … }

ω + 1 = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, …, {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, … }}

⋮

Moreover, every ordinal is a pure set, since there is nothing inside them other than the empty set and the onion layers of brackets.

The link between ordinals and well-foundedness implies what is called the Principle of Unique ∈-Minimality for Ordinals. Badiou expounds this principle in the first appendix to Being and Event. ‘If there exists an ordinal which possesses a given property, there exists a smallest ordinal which has that property: it possesses the property, but the smaller ordinals, those that belong to it, do not’ (BE, 519). Since every ordinal is well-founded when ordered according to belonging, every set of ordinals has a first element, the smallest ordinal of that set. This means that any collection of ordinals defined under some predicate has a unique first element.

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