Continuing with the summary of the video, from 54m to 1h08m.
According to Badiou this investigation is of much mathematical interest because it is an exploration from the inside of the characteristics of the Absolute.
This Absolute cannot be « injected » into the theory because it is the place of the theory, it is not itself a set. We can work on interior attributes that though they too are not sets are more approachable because they are defined by a particular property (e.g. « is an ordinal number », or « is a cardinal number », etc.) that is very general but specifies something in the Absolute.
The Absolute can thus be said to express itself inside itself, immanently, not only at the level of quantitative immanence (the class is inside it, and in that sense « smaller »), but at the level of the truths that it deploys, i.e. in the effective definition of the properties of a multiplicity. The Absolute is thus repeated in the attribute but also displaced, because there will be cases when j (x) ≠ x. So the attribute represents the Absolute from the inside while differing at the level of truths.
This means that the theory of the general forms of the multiple can admit correlations or internal movements which slightly displace a particular characteristic of the truths from the point of view of their application.
For example in Spinoza we can affirm that extension and thought are the same, because they have the same order-structure, but this does not explain how they are also different. It gives us a structural identity, without explaining how a thing and an idea are different. But in Badiou’s schema we have an explanation. It explains how the truths concerning a form of the multiple can manifest themselves in an attribute, in integrating a principle of differentiation at the same time as a principle of identity. One may say that this schema is a dialectical version of Spinoza’s attribute.
This takes care of Substance, attributes, and modes (elements x and j (x) are modes), but what about infinite modes? Here there is an absolutely magnificent theorem. We cannot demonstrate with the axioms of set theory that attributes of the Absolute Place exist. So perhaps there is only V the Absolute Place, perhaps it never expresses itself in different attributes. However, as we have seen this creates a problem for distinguishing thought and extension.
The theorem states that if there exists a non-trivial elementary embedding, then there exists a new previously unknown type of infinite cardinal, larger than all those we have defined up to now, called a measurable cardinal, i.e. there exists what mathematicians call a « witness » to the existence of attributes. Inversely, it can be demonstrated that if such a cardinal exists, then there exists a non-trivial elementary embedding.
So it has been demonstrated mathematically, which in Spinoza remained at the level of a brilliant but vague idea, that there is a constitutive link between something infinite, and the existence of attributes.
This theorem is one of the great masterpieces of modern set theory. It shows that from a structural hypothesis (the existence of a correlation internal to the Absolute Place under the form of an attribute), if it is valid, one can deduce an existential witness, i.e. the effective existence of a form of the multiple previously unknown.
This is an admirable political theorem. If you manage to think or to edify, inside or on the surface of the world as it is, something that is the possibility of expressing it, but within a difference, because j (x) is not always equal to x, if you manage to introduce into the world something that does not destroy it but which preserves it with a difference, and this difference can be very great, if you really do something without destroying, which is neither in the figure of destruction nor in that of simple preservation, something which is a differentiated internal expression, then a masterpiece will bear witness to it, i.e. a new existential infinite, a creation, something that did not previously exist.
So it establishes the political (in the widest sense) link between the structural actions of politics, i.e. its effective deployment in the possible dimension of an attribute of what is, and something which is a witness of this effort, something which is an existence of an unprecedented intensity. The reverse is also true, for the mathematician. If you are in the feeling of an unprecedented intensity then there is probably somewhere a structural effect that is both expressive and differentiated.
All this, Spinoza together with modern mathematics, teaches us the extreme value of the notion of attribute. That is to say, one must not allow oneself to be confined within the idea that there is only the alternative between conservation and destruction.
The attribute is an expression of the absoluteness of Substance totally differentiated from every other. Spinoza insists on this, he affirms that two attributes have nothing in common. Even if we are more prudent, and we maintain that an elementary embedding produces more localised differentiations, nonetheless it is a matter of difference. If you have the testimony of the existence of a form of life of an unprecedented intensity, K the giant cardinal, then it is the case that this structural modification has occurred.
There is a link of witnessing, and so a subjective link, between that which attempts to express absoluteness inside an uncontrolled situation and the creation or appearing of an unprecedented form of intensity.