13. Axiom Schema of Replacement
Infinite being is analogous. It does not emerge out of, but reaches across to.
Aphorism: Infinite being is analogous. It does not emerge out of, but reaches across to.
Advice: Reach out to other infinite beings and be transformed!
The lack of contradiction in sets due to separation, allows for an alternative version of the Axiom of Subsets. This is called the Axiom Schema of Replacement which we will take as a different axiom.
Many see replacement as a stronger example of how to form new sets than via subsets alone. Instead of generating sets from the inside, via subsets, you can map out a new set by replacing every element in the first set with elements of the new set, as long as you do it in order, 5th for 5th, 4th for 4th, and so on. And as long as the new elements already exist elsewhere in an already collected set.
Replacement creates at least two ontological insights. First, it helps us to appreciate how identity and difference are suspended in sets because they are radically altered (indifferential suspension). A set of 5 cats can be replaced by a set of 5 dogs, as long as you replace the 5th cat with the 5th dog, and so on. Two sets containing the same elements are identical, so two sets of 5 elements are identical, even if their content is different.
Here identity is not defined by characterisation of classifying entities, or with totalities which we can arrive at as a result. Identity is not really about being a collected multiple as such. It assumes collection, and it assumes other collections, and tries to identify them with each other using functions, transformation or mappings (Tiles 1989, pp.127-8).
A second value is that the Axiom of Replacement defines this functional mapping as opposed to classification of entities. This is important for mathematics which has been increasingly concerned with functions since at least the 17th century. It is also important for ontology.
For example, replacement allows for the formation of sets at the infinite level beyond ω + n, where n is a finite number. This is an important step towards the transfinite universe we all exist in.
It also helps to describe the truth of the situation of infinite existence which is, that being is not defined by what is contained in a set in its totality, but how infinite worlds map onto each other. As in a cardinal set, we do not know the size of the set, so it is impossible to describe the set fully using the conception of totals. The generic multiples of a cardinally infinite situation can only be described by forming a finite set out elements of infinite sets by mapping them functionally.
This will be the basis of the Axiom of Choice below. It also explains how selection works. The Axiom of Replacement creates infinite sets out of finite functions of existing infinite sets. For example +n where n is a finite number. It does so through mapping, rather than through the concentric, alliumatic, recursive, tabularity (CART) we described with the Growth and Emergence Axioms.
What we learn from the replacement is that at the transfinite level, where pairs of indifferent multiples are moving towards pair-potential generic multiples, sets grow in a very different manner.
Infinite sets no longer emerge from concentric embeddings, but are created by functional mapping. Being is not actually emergence, it is transferral. Infinite existence does not come out of, but reaches across to.
This is evidenced in the generic multiple. Of this multiple we do not know enough information. This is called by Hegel and Deleuze the indifference of indetermination. To fully know this multiple we have to functionally map its elements onto another set about which we know more. This multiple is the next smallest multiple. Then again, we know very little about this multiple also, it is similarly generic after all, so again we need to replace its elements with those of a smaller set, and so on.
This is the undoing or unravelling of immanence. It is the plotting back the levels of Russell’s ramifications. It is the retroactive stepping down of a hypothesis. Hypothesis means literally the founding of one thesis on another thesis. You are endlessly descended the stairs towards foundation. Like those images by Escher. Like Duchamp’s “Nude Descending a Staircase No. 2”.
The generic multiple goes downstairs.
Existence is a retroactive zig-zag, the chevrons of the stairway.
Here the recursive function is such that eventually we will arrive at the only pair we know information about, which is the first pairing of the multiple set. About this pair, for example, we know that it does not succeed. Yet, thanks to the Axiom of Replacement, it is consistent because it has been mapped functionally from an infinite set that itself is merely ω+ n with n being a finite number.
Travelling backwards, using functional mapping, we will arrive at the first, indivisible, infinite multiple: infinity or w. Infinity is at the bottom of the stairway to the transfinite paradise, not at the top.
ω (infinity) is a non-successor or limit ordinal, which therefore grounds our generic multiple, making up for the fact that it has no defined cardinality from which to count back from.
We don’t know all the information needed to stabilise a multiple, but we know its halting point and its relation to another multiple. Two out of three is not bad. It turns out it is enough.
The functional mapping mode of forming new sets out of old, could be the means wherein the largest set of finite numbers becomes the infinite. An infinite set is a set composed of already existing sets. It assures the existence of any set to which all the values of the unions of sets belong. This allows us to state that there is a set to which all the natural numbers belong. This is the union of all numbers, which is all the numbers +1, or infinity (ω).
Similarly, working backwards we can also say of the next largest infinite number, w+1 that it is a replacement for w, thus determining non-succession across the divide that is instilled by the Non-Continuum Hypothesis.
It is Badiou who first realises that the Axiom Schema of Replacement is an essential axiom for indifferentiation. He says that the axiom shows that the consistency of any multiple “does not depend upon the particular multiples whose chosen multiple it is” (Badiou 2005, p.65).
Thus extension allows for transfinite numbers, it explains the nature of the generic multiple, it removes the need for continuum, it replaces the classification and totalities of entities with their functions and transformations, and finally, it explains contentlessness. It is an essential schema for understanding infinite existence.
There is more. Because you can replace all the elements of one set, with all those of another, and it be the same set, because the content of the multiples is irrelevant (contentlessness), then the idea of identity is permanently destabilised. Or better, rendered inoperative.
What a multiple is, is where it is, not what it is. And where a multiple is, is not to do with where it is contained, but what it is capable of doing as a function.
If we now think of our generic multiples, it is not so much where the multiple is, first or second position, that matters, but what this allows the multiple to do functionally.
Normal pairs are insular and parochial. Generic pairs, in contrast, are cosmopolitan, outward facing, gregarious, and adventurous.
If a multiple belongs, it means it exists already and so can include other multiples to move towards a transfinite situation. If a multiple is included, that means it is the result of a prior mapping such that we can say it is part of the foundation of the consistency of the transfinite set.
Sets are promiscuous. Sets swap multiples with each other happily, constantly. Generic pairs are not bourgeois. They refute normative models of fidelity.
The multiples you claim to include, are not in your set because you collected them, necessarily. They are much more likely to have arrived in your set through a multiple swap facilitated by a shared function.
Sets can appear to be claustrophobically composed. They are mathematically dense. They appear solipsistic and inward looking. Especially when you compose finite selections using CART.
Again it is important to remember that we exist primarily as infinite sets. Infinite sets grow in a completely different way. They are not arithmetical, but functional. Theirs is not the mathematics of numbers, but of functions. Existence is an infinite doing. This is how we break free from the claustrophobia of separation.
There are two kinds of separation. One grows up from what there already is. The other leaps across from what it is, to what it is not.
Badiou himself makes a difference between two kinds of infinite collections, those determined according to sets, and those according to categories, or functions of relation (Badiou 2014). We have called this functional mapping communicability. It will form a major part of a future, indifferential epistemology (Watkin 2022).
Sets tell us what a multiple is composed of. They open up the set, to show the multiples. Categories, relational functions, tell us about the relationality between sets. They explain how one set acts on another, irrespective of content.
This is another way of neutralising the content of any set. It is also an important part of existentialism.
To what degree do we care what we are composed of? Are we not rather interested in what our functions are, as a composed being, and their effect on other composed beings?
Imagine an ethics of action, instead of a politics of identity. Be judged on how you collect, not on how you have been collected.
Historically, with Aristotelian classes, it was the combination of essence, judgement, and property, that determined relation. Beings were relational based on composition and quality.
In terms of replacement, two beings are defined as identical, two examples of the same being, because the multiples of one, can be mapped functionally onto the existing multiples of another. Although the function is size, which suggests composition, as we know, at the infinite level, size is a mode of describing an infinity. To describe an infinity is actually to map a recursive function from one set onto another. As I have said elsewhere, the function of size, represented by the relational symbol >, is only one of any number of functions that can be mapped across sets (Watkin 2020a).
We have become obsessed with size, when in reality the size of the set is irrelevant.
Bijection is a functional mapping of pairing.
We know that the set of all even numbers is half the size of the set of natural numbers if we speak of number in terms of totalities. Yet we also know that the function of pairing allows us to map all the numbers in the natural number set, onto an already existing set, all the even natural numbers. The result is a new infinite set. Bijection is the application of replacement using the function ‘to pair off’.
Diagonalisation is another functional mapping of rational numbers onto natural numbers. While diagonalisation’s failure to map real numbers onto the natural numbers 1-to-1, is the basis of cardinal infinities, as we saw.
Relational mapping usually has a domain which contains all numbers, and a codomain which is all the numbers you could map a function onto. Diagonalisation of real numbers shows that the domain, real numbers, is larger than the codomain, all natural numbers.
It is very normal that not all of a codomain is mapped, but it seems unusual when the codomain is smaller than the domain. We end up with this formula:
What this describes is: there exists a function between x and another multiple, Axiom of Replacement. This multiple must exist. But this multiple is not included in the infinite set of natural numbers. Thus other kinds of numbers must exist because this map exists.
Admittedly, most set theory appears concerned with finitude. The axioms of growth and emergence, for example, grow sets, but they do not grow them to infinity. This is due to the failure of the Continuum Hypothesis.
It is the Axiom of Replacement that better describes the growth of infinite sets. Growing sets using finite functions like +n. But also creating new kinds of sets using functions like the bijection of real numbers onto natural numbers. Diagonalised rather than linear growth.
The quickest way to grow your universe is to create new things, not to just keep adding numbers to the sets you already have. Replacement creates new numbers, new kinds of numbers, called cardinal infinities.
If our contention is that human lived experience is based on our cardinally infinite existence, then the Axiom of Replacement is the explanation of how we do this. We are not collected beings, at the infinite level, so much as functionally related, mapped, or analogical beings.
Being is analogy.
Analogy is the application of a rule or function, to another being to one side of you. Existence is this side-by-side relationality. We exist side-by-side due to 1-to-1 bijection. We exist alongside others. We also exist alongside ourselves (Watkin 2020b).
It would appear to me that ranking as counting is neither analogical nor functional. It can of course be described in a function, that of +1, but only at the finite level. We do not exist on that level without an intervention of derived selection. Another function which means finite existence is a functional mapping of infinity. Analogous sets are not, however, ranked. They are equal in size.
Replacement is the liberation of beings from the strictures of being ranked. Here are two pictures of modes of being.
Figure 10.
The first image is a picture of ranking. The second is a picture of mapping. They could not be more dissimilar.
The second diagram is a very good picture of generic multiples in an infinite being.
Let us take D as any multiple whatsoever in a finite set. We map D onto a chosen multiple from a generic set. We can say that D has a cardinality. It is a 5. We cannot say the same of R, because cardinal infinities have no cardinality.
How can we rank R or generic Rank therefore? Very simply. We map the function of Ranked from D, where this is clear (5 is ranked between 4 and 6), onto R where it is not. R behaves like any R. It is between two multiples, one smaller, one larger.
Rank, we realise, is simply that. The functional positionality of either being before or after the next nearest multiple. This is how generic multiples exist. All of this is possible without considering quantity, composition or totalities.
Being is analogy, existence is alongside. Finite being emerges from out of the pond and slowly finds its legs. Infinite being leaps like a Robert Lowell’s salmon.
O to break loose, like the chinook
salmon jumping and falling back,
nosing up to the impossible
stone and bone-crushing waterfall” (Lowell, “Waking Early Sunday Morning” 1967).
Infinite existence is the breaking loose, the jump and backwards fall, the leap up and across, the impossible.
Being is diagonal. It maps the trajectory of the leaping fish, as it tries to rise above the barrier of vertical rank.