The Set of All Sets
August 01, 2016
This article is about working out whether there is a God; by trying out explanations applied to everyday problems.
Some things we speak of don’t make sense when we look at them reasonably. Like the description of an experience as having been ‘bittersweet’ for us. If we chop this expression up logically, we know that sweet and bitter are opposing sensations of taste. We know usually a taste is either bitter or it is sweet; and that when we add sweetener to bitterness it will ameliorate that bitterness; and likewise when we add bitterness to sweetness the sweetness is toned down. The two tastes are not normally tasted together in the same mouthful as being both present and yet distinctively separate as sweet and bitter tastes.
In science, for those of you who did some simple chemistry at school, you learn about acids and alkalis; and about the litmus test for identifying them, one from another. Litmus paper turns red in an acid but blue to violet when put in an alkali. When you add acid to alkali in equal proportions the resultant item is neither acid nor alkaline; but neutral.
In this kind of sense to say a memory is ‘bittersweet’ in general does not tally with our experiences of mouth tastes and neither is it analogous to simple chemistry; ‘bittersweet’ here used is a distinctly psychological experience as opposed to a logical one.
So, what of it?
The example of ‘bittersweet’ was given by me just to demonstrate that logic and our expectations of what logic as able to and can do are not always straightforward, obvious or self-evident.
‘Bittersweet’ is called in grammar an ‘oxymoron’. Other examples of oxymorons are ‘open secret’ and ‘living dead’. Oxymorons contradict themselves; but in another way of looking at them they have due context and graspable meaning in the right situations.
And logic is not so daunting as all that; so don’t be put off. Aristotle gave us three basic laws of logic
- A thing cannot be in more than one place at one time
- A thing cannot be and not be
- A thing cannot occupy the same space that another thing is occupying Pretty simple stuff eh?
There are some more modern ideas hard to get one’s head around though; for instance those ideas to do with Quantum Physics which accept that a quantum of sub-atomic material is, or behaves like, both a wave and a particle. There is also an idea that two objects travelling fast enough and far enough away from and back to one another demonstrably can be shown to have somehow experienced different time durations elapsed. This is to say; one object will be older or younger than the other compared to how old each was before they parted.
These ideas don’t make sense; and maybe they are not sense? But they are ideas which scientists accept right now; and the scientists accept them simply because accepting them allows them to make Personal Computers and to go to the Planets, and such.
There is in theories of logic a thing called The Set of All Sets. You know that libraries catalogue their books – by subject area; by author; by title; by date published; by publisher - and so on – it’s all quite boring and only autistic librarians enjoy it.
But the cataloguing puts the books into sets of all kinds, so that a person coming in looking for, say, The ABC of Reading by Ezra Pound is able quickly to locate it on a shelf. Some libraries, like The British Library in London hold millions of books; and this careful cataloguing allows any one of them to be retrieved for a reader in minutes.
So sets are useful. Now have you heard of Russian Dolls or Chinese Boxes? They are kind of ‘toys’ where one thing fits inside another; dolls within dolls and boxes within boxes – up to three or four at a time usually but there are extreme items with double figures of boxes or of dolls. Sets can be arranged like the boxes and the dolls are arranged. Take a kitchen fork for instance:
All forks can be placed in a box marked ‘cutlery’; all cutlery can be placed in a box marked ‘kitchenware’; all kitchenware’ can be placed in a box marked ‘everyday utensils’; all everyday utensils can be boxed in a box ‘man-made things’, and so on.
But all forks can also be placed in a box marked ‘metal things’; all metal things can be placed in a box marked ‘things made of ores’; all things made of ores can be placed in a box named ‘natural resources’; and so on.
So cataloguing can be complicated; and some boxes will have more than one box visible inside them and these boxes inside might be seen to hold more or less other boxes.
But logic tells us there must be - in the catalogue of all things that are in existence - a set which is The Set of All Sets; that is a big box containing everything in it.
Now the crafty masters of logic ask the $64K question: Is this Set of All Sets in the Set of itself? This is a mind bender; the ultimate oxymoron; and a classic dilemma.
Because if the Set of All Sets IS in the Set of Itself; then it IS NOT The (one unique) Set of All Sets because it is only a component of a Set. If it IS NOT in The Set of All Sets then it is not The Set of All Sets because it belongs to a Set other than this Set of All Sets.
This paradox holds true so long as every object is able to be catalogued within a set (except of course The Set of All Sets).
What we get now if we try a simple way to solve the paradox; by putting our previous Set of All Sets in a bigger box and calling that bigger box The Set of All Sets; is called ‘infinite regression’. When my boys were young they watched a video cartoon in which two daft workers had to fill a hole in a road surface. The two workers dug another hole in the road and filled the first hole with the dirt. Then another and another and so on; wondering why they always ended up with an unfilled hole to fill!
Well that is infinite regression and the new bigger box to hold the Set of All Sets suffers the same paradox as the former smaller box and no matter how big a box and how many times you add one to the problem it will always be the case.
The problem is simple enough. The guy who first put this paradox forward used a variant on it to show (to his own satisfaction) that if one proposes that there is God; one has to propose that there must be a greater thing than God - into infinite regression.
This guy just did not abandon logic when logic failed and became unusable. He just left it there as a paradox and a problem; as an oxymoron and a dilemma.
There are plenty of other blocked routes like this that show themselves when one looks into trying to think through by using reason alone and without allowing concessions to the reality that some things are present in life which are not solvable or graspable by reasoning alone.
And one has also to grasp in addition to accepting the limits of logic and deductive reasoning; that it is not any use to batter one’s head against obviously self-evidently insoluble logical conundrums. There is no need to do so. Just as ‘bittersweet’ as a word with meaning resolves itself wholly psychologically in our minds when we use it to describe an emotional experience; and we indeed do so, then we are able in the same way to consider a possibility of a God; without experiencing confusion and without sensing a contradiction or problem .
We are able also to accept God in the same way and with the same equanimity of mind as we accept ‘bittersweet’ or ‘open secret’. Our understanding of God, as this is aided and added to by a reading of the Gospels, offers us ample confirmation that we are ‘allowed to’ do so; and that it is permissible to think in these terms; and that we do not have to force everything we think and believe through a logical straw or strainer every time. Whom we are; what we think - do not have to pass any rigid empirical feasibly and logic test.
Take these sayings of The Lord Jesus; which are all paradoxes, yet are they also all deep incontrovertible spiritual truths:
‘If one wants to be great among you, he must be the servant of all’ ‘Your brother was dead, but now he is alive’ ‘From whom much is given much is expected’ ‘As you measure out, so shall you be measured out to’ ‘It is better to give than to receive’ ‘He who loses his life for my sake shall find it" ‘With humankind it is impossible, but with God all things are possible.’ ‘The first shall be last and the last first’ ‘Love your enemies’. ‘If a man strikes you on one cheek, turn to him the other cheek’
I have over a long period of time come to accept life as the great paradox; and to embrace a life in pursuit of and in emulation of Our Lord Jesus; who ‘saw into the heart of things’ and there is he present ineffably.