Consider an alphabet made up of symbols, each of which is a square of side length 1cm that is black below a line at height r cm above the base and white above that, where r can be any real number in the interval (0,1). Then the alphabet has the same uncountable cardinality as the real numbers, and a one-to-one map between the symbols and the reals is that which maps the symbol with line height r to the real number tan((r - 0.5) x pi / 2). — andrewk

Neat example.

See what fruits come from simply accepting wittgenstein and david stove as your therapists and saviors . Be addled no more by the barbs of idle thought!

It's more entertaining than doing crosswords.

i take offense to that as a long time crossword fanatic. but cmon youd 'psychoceramic' this thread without two blinks if it was someone else.

anyway it just goes to strengthen my a j ayer theory:


The most 'sensible' among us are sitting atop a reservoir of wild shit

i take offense to that as a long time crossword fanatic. — csalisbury

Suck it up.

youd 'psychoceramic' this thread — csalisbury

Nu. It doesn't espouse a wide enough explanation to count as a cracked pot. The stuff I wrote this morning about the role of the Awesome in education, now that's much more terracotta.

Just to add...any given list of proper names of actual people will have duplicates...triplicates...Like Socrates the footballer and Socrates the philosopher...the many Kims of Korea...the andrewks and mcdoodles...

So where will all this counting of the different names get us?

If you like. That point is irrelevant to a discussion of the use of constants in first order language. @andrewk's example above appears clear enough; it's not about people.

Andrew seems the only one who can see the interest in an esoteric bit of logic. That's a bit sad.

but thanks, Andrew, for your help.

Consider an alphabet made up of symbols, each of which is a square of side length 1cm that is black below a line at height r cm above the base and white above that, where r can be any real number in the interval (0,1). Then the alphabet has the same uncountable cardinality as the real numbers, and a one-to-one map between the symbols and the reals is that which maps the symbol with line height r to the real number tan((r - 0.5) x pi / 2). — andrewk

What about pre-writing? Are spoken proper names countable?

Why?

Why what?

why ask such a question?

What about pre-writing? Are spoken proper names countable? — Michael

Better to focus on audible names, rather than spoken names, in order to transcend the limitations of the human larynx.

Audible names need not be countable. We could generate an uncountable set of names as follows. Let every name be a sound of length two seconds, that is a pure tone (sine wave) of frequency 800Hz and constant amplitude. We could map the written symbol that is the square whose black part has a height to width ratio r, to a tone with amplitude A + r (B - A), where A and B are widely different amplitudes that are both within the comfortable range of hearing of most humans. Then we distinguish sound symbols by amplitude, and we have an uncountable set of amplitudes from which to choose - the numbers in the interval (0,1).

Just noticed this. noice.

Jim, Jeff, Jenny... that's three.

1 is the proper name for that number; 2 , for the next number. and on it goes. So there are at least countably infinite proper names.

Suppose I take the set of infinite lists of ones and zeros. I know from Cantor's diagonal that this is uncountable. So I give each its own name; are there then uncountably many proper names?

First-order predicate logic apparently assumes only a countable number of proper names: a,b,c...

How would it change if there were an uncountable number of proper names? — Banno

I suppose I'd prefer to distinguish the number of proper names in a given universe from the number of proper names indicated by, say, a formal notational system of predicate logic. I'm not sure which of these you're question is aimed at. I suspect it may be a question about the notational system.

If it is not, then I suppose the answer must vary along with the universe given, and that there is no satisfying answer to the question in general.


Each proper name, I recall vaguely, is a unique logical identifier for each particular entity recognized in a logical universe, so there is a one-to-one correspondence between entities and proper names in a universe. There may be many entities in the same universe called Tom Jones; accordingly, a name like "Tom Jones" is not the logically proper name of any entity. Each thing that exists as a logical object in a predicate system has its own proper name. Is that about right?

In that case it seems the proper name is a sort of logician's posit or fiction or theoretical construct. How many of these are there in a given logical universe? As many as the logician who constructs the universe pleases.

What is the total number of logical universes actually constructed by actual logicians in the course of the actual universe; and how many proper names did each of those logical universes in fact contain? I suppose that's a sort of empirical question.

Suppose I take the set of infinite lists of ones and zeros. I know from Cantor's diagonal that this is uncountable. So I give each its own name; are there then uncountably many proper names?

First-order predicate logic apparently assumes only a countable number of proper names: a,b,c...

How would it change if there were an uncountable number of proper names? — Banno

The point is that logic derivations have to be of finite length. So you can never use more than a finite set of names in a formal proof. Even if you imagine to have an uncountable set of names, the set of names that you can use in any derivation, however complex, will always be countable. So, it doesn't make any difference what's the cardinality of set of names that you have. The only thing that counts is the cardinality of the set of names that you can use in a derivation.

You're supposing that there somehow are proper names that no one has said or thought?

No, names used in logic are simply strings of characters: formal logic is a purely syntactic system.

You could imagine to use geometric objects (whose sizes are supposed to be an uncountable set) instead of strings for names, and use geometric constructions as rules. In that case you would have a "logic" based on an uncountably infinite set of "names", but then the problem of recognizing if two names (or lengths) are the same I think would become undecidable: there is no physical way to compare an uncountable set of lengths to decide if they are the same.

Perhaps I'm missing something about uncountable sets. Can one set with aleph-1 elements be mapped to another set with aleph-1 elements?

So could an uncountable number of individuals be mapped to an uncountable number of names? — Banno

Every mathematical object has a proper model of itself.... basically itself. So basically (not rigorously) it means that R=R

Proper names only occur when someone thinks or says one.

In this case surely there are proper names that no one has said or thought: just take a random string of 30 letters (and maybe add some vowels to make easier to pronounce): there is a very high probability that nobody has ever thought or said that name before!

Proper names only occur when someone thinks or says one. — Terrapin Station

But that's not right. "Two" is a proper name. Same for any integer. There are integers that have never been thought or said. Hence there are proper names that have never been thought or said.

This is what I had in mind. It's easy to make such a mapping for countably infinite stuff.

What about uncountably infinite stuff?

Hnece:

Only a countable number of objects can be referred to individually, because there is only a countable number of names (aka 'constant expressions') that can be used to refer to them. — andrewk

and

It is readily proven that for any positive integer n, there is only a countable number of different n-tuples from a countable alphabet. One proves this by constructing a one-to-one map from the positive integers (which are countable) to the set of all such n-tuples.

It is also readily proven that a countable union of countable sets is countable (it is in fact the proof from the previous paragraph, applied to the case n=2). The set of all finite strings from a countable alphabet is the union, for n going over all positive integers, of the set of all strings of length n, each of which we know is countable from the previous paragraph. This is a countable union of countable sets, and hence countable. — andrewk

Which seems to settle the question in the negative.

And how is it a proper name prior to being used as such?

There are integers that have never been thought or said. — Banno

There are? Where? And how do their names exist prior to being named?

Maybe I didn't understand what you mean by a "name". I was thinking about names used in logic propositions, that are simply meaningless labels.

"Proper name" aka "proper noun": "A noun that is used to denote a particular person, place, or thing, as Lincoln, Sarah, Pittsburgh, and Carnegie Hall." (https://www.dictionary.com/browse/proper-name)

We could extend it to "names" (do you mean variables?) used in logical propositions if you like. How are there any of those if someone didn't think or say them?

Well, for proper names that denote a particular object, I think somebody must have assigned a name to the object before you can use it, so the answer is no. For variables of logical propositions, they are only arbitrary strings or arbitrary length (at least in formal logic), so there is no need that somebody assigned a meaning to the name before using it.

For variables of logical propositions, they are only arbitrary strings or arbitrary length — Mephist

That exist where/how prior to someone (or something, like a computer) creating/assigning them?

OK, now I see what you mean. You are saying that maybe there is no meaning in saying that a particular string of characters, or a particular number "exists", if nobody has never written or thought about it in some way. Well, I am convinced Platonist. I do think that all possible numbers, or strings, "exist" in some concrete sense, even if nobody ever thought about them, or even when human beings didn't exist yet on earth.