Lawvere wrote in a couple papers that Cantors word “menge” which is usually understood as “set” is actually a cohesive type. And the “kardinale” is the abstraction from this by getting rid of the qualitative aspects of this “whole?” and what you get is a bag of points serving as a universal number was.

What I don’t get is that Cantors description of it seems to be sort of contradictory. Cantor says that he feels that numbers are like the forms of sets. That the number abstracted are actually more United then the whole they are abstracted from.

This seems contradictory to me. Unity seems to be a passage towards a nondual or monistic object. Whatever your conception of it might be. The plurality would imply dicreteness, no? Although cantor never said how points that are homogenous can be distinct. Maybe Nishida’s absolute nothingness concept is an answer. Where he nondual is the ultimate differentia?

I digress. The unity of these numbers seem to me to be the actual separation. His language also expresses an Aristotelian tone where form and matter play a role. Cantor describes the points as the matter and the order underneath as form.

Prime matter however is the potentiality, it is formless and does not exist. Is this nishidas absolute nothingness? If not, then the prime matter seems to be the continuous in Aristotle’s conception and platos ideas are more like discrete distinct forms.

Cantor seems to believe numbers are multitudes of units. These units are unity but the multiplicity of them would imply a separation and discreetness introduced to it, no?

Does Cantor believe quantity comes before quality? Can we not have a system like Hegels where quality is actually first?

I’ve thought that maybe he is referring to prime matter as formless as the “menge” and as the number as the forms. But then how is quality explained? Just differing manifestations or attributes that this number abstracted things may become?

The way he describes menge however is a bunch of distinct thing considered as a whole. The prime matter is not distinctive to anything or it’s both and not at the same time. It is pure potentiality, and Cantors notion only seems to make sense when we consider consciousness as prime matter or the unity “of our consideration”.

If numbers are forms and they seem to be abstracted from the menge which were whole without qualities and then had qualities and then we abstract all qualities away then wouldn’t number he form without matter?

Would not form without matter just equal matter without form?

How can a set be understood as a cohesion in any sense when elements are so important? Oftentimes in set theory we only talk about elements and simply ignore the background which allow them to “register” as something with just a capital letter label. In fact, we start with an empty set and singleton set.

What is this original menge that cantor describes? Is it a continuum with differing qualities and each qualities boundaries are those which tell us that this boundary is a “unit” or element?

Any insights would help. Google translate is not very friendly to the German language as well it seems.

  • 2
    "How can a set be understood as a cohesion in any sense when elements are so important?" The gist of set theories is exactly this: a collection of objects is a set that in turn is an objects in its own. See B.Russell, PoM, § 127: "a class seems to be not many terms, but to be itself a single term, even when many terms are members of the class." – Mauro ALLEGRANZA Nov 5 '20 at 8:53
  • 1
    Lawvere is co-author of a full book on set theory: FWilliam Lawvere & Robert Rosebrugh, Sets for mathematics (Cambridge University Press, 2003) that "rebuild" set theory from the viewpoint of categories. – Mauro ALLEGRANZA Nov 5 '20 at 10:11
  • 1
    Are you referring to Cohesive Toposes and Cantor's 'iaufcer Einsen' by Lawvere? And could you ask something more pointed than "what is the connection" and "any insights would help"? Lawvere already wrote a whole article on the connection. – Conifold Nov 6 '20 at 22:27
  • 1
    @Conifold ... continued. If these abstract sets are universals then it seems cantor is giving a sort of privilege to these universals as opposed to the nature of things which he claims we need to abstract away to get to the cardinal. Why can’t “redness” remain from the abstraction? It seems that his conception of set is just a thing which has every possible thing and attribute and if we are to abstract from it we would necessarily arrive at the cardinal as the sort of bare minimum. But then, how is the cardinal 4 differed from 5 in this case? They are abstracted from the ultimate, no? – enrijaja Nov 6 '20 at 23:05
  • 2
    I do not follow most of this, it is hard to follow the post as well. Cantor first abstracts from any qualities of objects, other than for the purpose of distinguishing them, to get elements in his sets. He then further abstracts to hypostatize classes of equipollent sets into cardinalities. – Conifold Nov 6 '20 at 23:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.