I know I said I wouldn’t do a bunch of model stuff, but today’s idea really needs it. So, go back to the previous post and it’s pretty clear how Zermelo intended his axioms to be interpreted. Step by step building up of a larger and larger set theoretical universe with no top.
Just to go through it real quick, let 0 be the empty set and for any set a let z(a)1 be the set containing the set a. The axiom of infinity is that that there exists a set Zermelo which contains the empty set and the image of z over Zermelo is a subset of Zermelo.
But what is a subset? If a function is an injection, then its domain is a subset of its range2.
Most of the axioms of Zermelo-Frankel from the previous post are about subsets: separation, power set, union and choice. The idea of all of these is to thicken up the universe with enough functions that everything works how we intuit it - or how Cantor did at least.
Let’s give a concrete case. There is are two sets 2 and 3. The set 2 that contains all and only 0 and z(0). The set 3 that contains all and only 0, z(0) and z(z((0)). To say the set 2 is a subset of the set 3 is to say there is an injective function over 2 whose image is (part of) 3. For instance, the z function does this.
But - as Skolem realized - thickening the universe so functions work the way we expect is only one way to satisfy the axioms. One could also take the opposite approach of thinning the functions to just the ones that work.
For instance, add a set or function for each set existence or function existence proof in first order logic (which can basically only order up new sets a few at a time). Then you have at most countably many sets, a couple from each proof.
But one proof says there is an infinite uncountable set, call it Skolem, and one axiom (seperation) says you can get ‘all’ the subsets.
What gives? To say that Skolem is infinite and uncountable is to say there is one function (an injective function with Skolem as its domain and range which is not a surjection) and there is not another (a bijection between Zermelo and Skolem). All we have to do is erase the second function and Skolem becomes uncountable, regardless of how it is constructed. Just because Skolem is uncountable doesn’t mean there are enough functions lying around so that ‘all’ the subsets make an uncountable universe.
Working in this way, one can see that most of the properties of sets are relative to the model the set is conceived of operating in.
Let’s go through another example. A set is finite if and only if all injections from the set to itself are also surjections. This is a relative property.
Start with the sets 0, z(0), z(z(0)), etc.. We make a new set Robinson, which has the property each of the above sets has an injection onto Robinson. If you wanna be specific, let the injection be like z: f(0)= z(0), f(z(0))= z(z(0)), etc.. Finally, add to this system the rule that any injection from Robinson to itself is the identity3.
There’s one big difference between Robinson and Zermelo: Robinson is finite and Zermelo is infinite. But look at these boys in the good old misleading but intuitive list style:
Z={0,{0},{{0}},…}; R= {0,{0},{{0}},…}
Where is the difference? Zermelo and Robinson differ not in their local behavior but in their global behavior. They are in different contexts. If a universe has Robinson rather than Zermelo, then that universe is called “non-standard”.
Okay, so that was a long tour. Let’s get to the general philosophy lessons I promised. Well, they basically all have to do with relativism.
The first is the famous context principle of Frege. In this, as in so much, Frege followed Plato. When Socrates went on his quest to define a word, he never allowed his victim to simply assert “When I use a word … [the word] means just what I choose it to mean—neither more nor less.”. Meaning is use - disciplined use - and not intention.
The other question is one of the classics of contemporary philosophy. We looked at some weird sets, like Skolem, which are uncountable in a countable universe. From the inner perspective of the model, this is normal. But wouldn’t a human being ‘feel’ the missing subsets? Isn’t the mind unlike a formal model in this sense? This was famously proposed by Roger Penrose above.
This proposal is not as wacky as it at first seems. In many mathematical universes, all functions from the reals to the reals are uniformly continuous. An actual sound wave is always a smooth function of position. But audio engineers somehow intuit what square waves are. Mathematicians somehow found the universe with these “missing” noncontinuous waves long before the smooth ones, to the point that this is the universe that gets called “classical” and “standard”.
On the other hand, a process which is able to find missing objects should be able to calculate functions faster than a process that cannot. As emphasized by Thomas Hales, formal computational processes are actually faster than the informal social Hegelian process through which ordinary 20th century mathematics was distributed.
One can say these two points of view are not contradictory, as the Hegelian process is not a formal system so reasoning that applies within formal systems does to apply to it. But they certainly sit uncomfortably.
The last thing I want to write about is the famous question of what we can know about the divine. Is the creator good in the way a man is good, but to an enormous degree? The obvious problem with this is that this entails that a larger degree of good is possible. If the degree is large but finite, why not infinite? If infinite, why not a larger infinity?
This reasoning lead people like Maimonides above to propose what is called ‘negative theology’. In negative theology, all predicates applied to the creator - good, mighty, slow to anger, etc. - are regarded as metaphorical. This is why Maimonides places a huge orientation and high standards on textual interpretation.
But isn’t this a bit of an old fashioned view of what science is? After all, as emphasized by Hume and Husserl, empirical science doesn’t reveal whether objects obtain predicates, but how they sat in relation to an observer at some time. Martin Buber (pictured above) was emphatic on this point: “‘Then what do I experience of you?’ Nothing, for one does not experience You. ‘What then do I know of you?’ Only everything, for one no longer knows particulars.”.
This leads me to the last point I would like to make. Kant had a famous counter argument against the Ontological Argument found in Leibniz, Spinoza, Descartes and the scholastics before them, such as Anselm Of Canterbury above.
Deus is the supremely actual being, says Spinoza. Existence is not a predicate, says Kant.
Following Kant, Bertrand Russell and Poincaré emphasized the difference between properties which are predicative and those that are impredicative. In Russell’s examples, “Napoleon was Corsican” involves the predicative property ‘was Corsican’ and “Napoleon had all the qualities of a great general” involves the impredicative property of having all the positive properties a general. This second sentence seems to have a sense only if there is a set of positive properties a general, and yet it seems like it would be a great coincidence if could go from syntax of a sentence to an object like that.
Maybe this would be more clear if I used Frank Ramsey’s example, which is more or less “Carel is the tallest man in the room.”. This is unobjectionable only if there is there is an established meaning of ‘the room’.
Kant is certainly correct that to define God as the supremely actual being is impredicative. The definition defines God as a special member of a set of beings of which God is a member. This is unobjectionable as logic only if ontology has already been established. The ontological argument is subtly circular.
What is less clear is if this subtle circularity is fatal. In a similar way to Zermelo relativizing properties to an inductively built up universe of sets, Spinoza relativizes his logic to psychology. The property of ‘is God’ is that of a being who has no conceivable limits.
Spinoza’s God involves distinctions of a global nature, like the distinction between Zermelo and Robinson. From the point of view of ‘negative theology’, that this property is a relation in disguise might be seen as a positive rather than a negative.
Perhaps it was this psychological type hirearchy that Spinoza was thinking of when he said: “But if in any instance I found that a result obtained through my natural understanding was false, I should reckon myself fortunate, for I enjoy life, and try to spend it not in sorrow and sighing, but in peace, joy, and cheerfulness, ascending from time to time a step higher.”
Short for ‘Zermelo’s Successor Function’
This makes every, i.e., every one element set a subset of every two element set, but because it’s very rare for something to depend on the difference between isomorphic sets, I will just use “subset up to isomorphism”. Also, this definition works in weirder categories than set.
I’m not gonna worry right now whether this process actually results in a model of Zermelo Frankel set theory. For now, just let it be some arbitrary category.