(A) If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them bothThis is a rule (the ‘axiom of pairs’) that tells us that we can ‘construct’ a set {a, b} given the existence of its members a and b. We need this rule because we cannot infer the existence of a mathematical set, an individual object different from either of its two members, from the existence of its members alone. The consequent does not logically follow from the antecedent. In this, by contrast -

(B) If Peter preached in Jerusalem and Paul preached in Jerusalem, then Peter and Paul preached in Jerusalem.

we are not giving a rule for constructing any non-linguistic entity, nor are we making any existence assumptions beyond what is given in the antecedent. (B) simply gives a rule for constructing

*expressions*: it tells us that the consequent means the same thing as the antecedent. Given the propositions ‘Is_F(a) and Is_F(b) and Is_F(c) and …’ the rule allows us to construct the proposition ‘are_F(a and b and c and …)’.

So my question remains. We assume the following

(1) At least one element exists

(2) One element is finite

(3) Any finite x’s and a single element are finite

(4) Any finite x’s are such that there is some y such that y is not one of the x’s.

This does not ‘construct’ anything. Rather, it

*asserts the existence of*certain things. The only things it explicitly asserts are the existence of finite things. For example, it asserts the existence of one thing (the ‘first’ thing). It asserts (by inference) the existence of two things (the first thing plus some y which is not that thing), the existence of three things (the first two things and some other y), all of which are finite. The question is whether from statements 1-4 we can also implicitly infer the existence of infinite things (an infinite oset)

*in exactly the way that we can infer the existence of Peter and Paul from a statement about Peter and a statement about Paul*. Can we construct an

*expression*that refers to all of the elements of the domain? For if we can, it follows that all the elements of the domain exist – whether or not we actually constructed the expression. Peter and Paul exist whether or not we have an expression such as ‘Peter and Paul’. Do all the infinite elements of the domain exist, whether or not we construct the expression ‘all the elements of the domain’?

I hope this makes the problem clearer.

## 14 comments:

Hello Ed,

Yes, that does indeed make things clearer. But is this a difference that makes a difference? Rather than see your 1--4 as involving existence claims, let's see them as definitions or rules for using the word 'finite'. (4) can be seen as a rule for asserting the existence of an 'infinity':

From 'every finite x’s are such that there is some y such that y is not one of the x’s' infer 'there is an infinity of things.'

At this point a set-theorist would take the union over all these infinities and say that gives us 'all things'. But this is not open to us. My inclination is to claim that 'all things are' is interchangeable with 'every thing is' (*). We would have to go on with rules for using 'includes' and show that from 'there is an infinity' we can infer 'all things includes an infinity'. Needless to say I haven't worked this out in detail. Is there a reason for thinking that this would not work?

A justification for (*) might include the observation that we can do (and did) number theory perfectly well without the idea of 'the set of natural numbers'. Nowadays we say things like 'the integers form a group under addition' or 'the rationals are dense in the real line' but these can all be unpacked into statements involving 'every' and references to individuals. They look like non-distributive plural predications but aren't. Do we detect a difference between 'let n be an integer' and 'let n be one of the integers'?

>>At this point a set-theorist would take the union over all these infinities and say that gives us 'all things'.

Or the Axiom of infinity itself would do. This asserts the existence of a set containing all the finite sets. But no finite set can do this, ergo etc.

>>My inclination is to claim that 'all things are' is interchangeable with 'every thing is' (*).

So ‘every x such that x not in x’ is interchangeable with ‘all the x’s such that x not in x’? In set theory: {x: x not in x}.

>>We would have to go on with rules for using 'includes' and show that from 'there is an infinity' we can infer 'all things includes an infinity'. Needless to say I haven't worked this out in detail. Is there a reason for thinking that this would not work?

Well we could just take it as an unassailable principle that we can always speak of {x: Fx}. But that is unrestricted comprehension. IN Ockham theory will this lead to the same problem as in standard set theory.

Sorry Ed, don't follow you. The equivalent in Okhamese of set membership is the is-one-of-the-x's predicate. 'every x st x is not one-of-the-x' doesn't look terribly well-formed to me, though you haven't yet told us how you see oset comprehension working, if at all. This looks rather problematic: the n's such that n*n+bn+c=0 may turn out to be nothing at all, a singleton, or a two-element oset I guess. These three cases will have to be dealt with separately. The case analysis will look like early mathematics before negative numbers were admitted.

Do you see a difference between 'n is an integer' and 'n is one of the integers'?

> This does not ‘construct’ anything. Rather, it asserts the existence of certain things

This is angels-in-pinheads stuff. It is constructing things by another name. If you don't like the word "construction" I suppose we can rephrase all discourse ot avoid it. But you can't hide the fundamental fact that "forming" or "describing" an oset "groups" objects together. It will all get rather unwieldy if I have to not say "constructing".

> We need this rule because we cannot infer the existence of a mathematical set, an individual object different from either of its two members, from the existence of its members alone

Of course. Because the mathematical set doesn't exist because of the existence of the two elements. It is the grouping of the two elements together that makes it a set. You do the same with language, except less clearly and precisely.

1-3 are basics, 3 in particular is a rule for making new osets from old. 4 is a statement about your universe: it asserts that it is infinite. But if those are *all* your rules for "constructing" osets, then no: you cannot "construct" an infinite oset using them, only a sequence of finite ones. But so far you haven't escaped what std set theory would say in exactly the same circumstances.

> The question is whether from statements 1-4 we can also implicitly infer the existence of infinite things (an infinite oset)

No, you can't, unless you mean something hidden by the word "implicitly". If you mean "using 1-4, and other assumptions which I haven't told you" then the answer is "maybe, depending on the other assumptions".

In, on, you know what I mean.

It is the grouping of the two elements together that makes it a set. You do the same with language, except less clearly and precisely.I find this claim interesting, because it seems on the surface contrary to the actual facts of language, which can, but doesn't typically, proceed by identifying extensions, whether clearly and precisely, and then grouping them. What sort of reasoning underlies it?

>>The equivalent in Okhamese of set membership is the is-one-of-the-x's predicate. 'every x st x is not one-of-the-x' doesn't look terribly well-formed to me, though you haven't yet told us how you see oset comprehension working, if at all.

On the well-formed claim, you are right. On the second claim (about how oset comprehension might work, I don't know.

>>It is the grouping of the two elements together that makes it a set

No , no and absolutely no. Nothing is being 'grouped' except the languagey bits. What is the difference in meaning between

1. Peter preached in Jerusalem and Paul preached in Jerusalem.

2. Peter and Paul preached in Jerusalem.

Nothing whatsoever. The only grouping is in the expressions 'Peter' and 'Paul'.

>>You do the same with language, except less clearly and precisely.

The only imprecision is in the concept of 'grouping' real things. Please explain this. The grouping of the linguistic bits is very clear and precise. You take two separate singular propositions, and rearrange them into one proposition with a grammatically plural and a subject consisting of concatenated proper names. Nothing could be more precise.

>>This is angels-on-pinheads stuff.

If you don't understand the difference between saying that something exists, and constructing something, i.e. causing something to exist, or if you think it is trivial, then we are not going to get very far.

If I have an apple, and a pear, then those objects exist (weeeeellll. If you like. Of course they are just convenient labels for a large collections of atoms that stick together for a while). They are real. But the set of "an apple and a pair" is an idea in my mind.

> constructing something, i.e. causing something to exist

Depending no what you mean by "exist", that is wrong. I would say that arbitrary groupings don't exist in the real world, only in the mind (whatever that means). So constructing a set doesn't make that set "exist".

But we're in danger of spending ages just arguing about language. This is what maths gets you: you don't have to do any of that.

>>But the set of "an apple and a pair" is an idea in my mind.

What does ‘the idea of a set of an apple and a pair’ refer to then?

>>But we're in danger of spending ages just arguing about language. This is what maths gets you: you don't have to do any of that.

So much the worse for maths. Some of the other things you said in that post will cause jaws to drop, by the way. I’ll leave it to others to comment.

These two diagrams may help explain the difference between Ockham sets and mathematical sets, as I understand it. It's very much a case of

reference, which is depicted by the arrows. Linguistic terms are shown on the left, in quotes to emphasise their symbolic nature. The Ockham term 'a and b' refers directly to the two objects a and b. In contrast the mathematical term '{a,b}' refers to a 'set-object', depicted as a quiver containing two arrows pointing to the objects a and b. Thus the mset {a,b} is distinct from its members, a and b, though it refers to them. The empty set, {}, is not nothing. It's a quiver with no arrows. It's this indirection that enables msets not so much to contain other msets but toreferto other msets that give the mset language it's referential power. My view is that msets are an artificial extension to the referencing capabilities of natural language. This may explain why we feel that they have to be 'constructed'. Contrast with the Ockham world in which such intermediaries are absent.I love the diagrams. Yes.

I have always despised the use of diagrams and pictures. But this quite eloquently shows the clutter and mess left behind by mathematical set theory in contrast to the elegant desolation of Ockham world.

Well, they do say silence is golden. :-)

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