Ordinal numbers are a type of numbers which are used for ordering. — Metaphysician Undercover

That's kind of okay in a very informal sense. But, just to be clear, it is not the definition of 'ordinal'.

Ordering is what defines the "ordinal" aspect, not the "number" aspect. — Metaphysician Undercover

I don't know what you mean exactly by "the ordinal aspect" and "number aspect".

If R is a well ordering of S, then there is a unique ordinal L such that <L epsilon-ordering-on-L> is isomorphic with <S R>. That implies that the cardinality of S equals the cardinality of L.

Anyway, I don't know what point you're trying to make. You disagreed with what fishfry wrote, then he explained how your disagreement is incorrect. You seem not to understand his explanation, though it was eminently clear.

In this context, there are two senses of 'count':

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five."
— TonesInDeepFreeze

Right,

one is a verb signifying an action, the other is a noun, signifying the result of the action. — Metaphysician Undercover

'count' is also a verb. But here I am mentioning two nouns.

A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)).
— TonesInDeepFreeze

This is what I have been telling you is incorrect. — Metaphysician Undercover

You don't even know what I'm saying. You don't even have the mathematical vocabulary.
You have no standing to tell me what is incorrect in this matter.

A count does not imply an order. — Metaphysician Undercover

I showed you how it does. And less formally, even a child understands that when you count, there's the first item counted then the second item counted ...

you cannot define, or describe counting as ordering — Metaphysician Undercover

And I didn't.

you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour — Metaphysician Undercover

There's measuring and there's counting. A measurement might not itself be a (human) count. For example, a digital scale may measure the flour without a person actually counting. On the other hand, counting would be to count the marks on a scale up to the mark where the needle landed. Your argument is grasping at straws.

you can see that there are five books on the shelf without placing them in any order — Metaphysician Undercover

We're not talking about taking in at a glance a quantity. We're talking about counting. You're grasping at straws. I notice you tend to do that after a while in a thread.

On the contrary, sets have no inherent order.
— fishfry

Exactly what I've been arguing, a count is a quantity, not an order, — Metaphysician Undercover

count(2) is a number (quantity, if you like). count(1) is not a number or quantity. We're talking about count(1).

you remind me of the border guard
who demanded the TAO-TE-KING from Laotse. — Trestone

A better comparison is with the boy who declared, "The Emperor has no clothes."

one would have to conclude that there was manipulation involved. — spirit-salamander

Less than a perfect distribution would have you infer cheating? That's crazy. And even if there were an astronomically long streak, one would very highly suspect that there was cheating, but it's not certain that there was cheating. There will be streaks. There is no upper limit on how long a streak can be.

Purely mathematical probability is not taken necessarily to be matched every time by real world outcomes.

Take the simplest example of a coin toss. The chance of heads is 1/2. But that does not entail that heads comes up exactly 50% in every experiment. Take even just an experiment of two flips. Easily it can happen that heads comes up twice - 100% heads rather than 50% heads. Indeed the probability of heads twice in a row is 25%. The probability of heads six times in a row is1/(2^6). So if you see it happen, then you happened to be there when that 1/(2^6) chance was realized. Just ask any mother and father who had six girls in a row.

If a probability of 1/6 for the occurrence of the six does not ensure that the six occurs within 6 rolls, then the 6 could never occur. — spirit-salamander

Probability does not ensure that the six will occur, but that does not entail that probability ensures that the six will not occur.

For any statement Q, from 'not ensured that Q' we do not infer 'ensured that not Q'.

It would not be ensured that the six occurs in 60 throws, not in 600, not in 6000, and so on. — spirit-salamander

Correct.

But what is the point of using probability if it is not reliable? — spirit-salamander

It's imperfect for making predications, but it's still good.

We don't need infinitistic considerations to see that the gambler's fallacy is incorrect.

The conditional probability, no matter the results of any finite number of previous trials, is the same as the initial probability, which is 1/2.

Inductive arguments can not show their conclusion to be true — forrest-sounds

True, inductive reasoning is not deductively valid. But inductive reasoning is a different form from deductive reasoning and has its own standards.

[1/6] can only mean that out of 6 times rolling the dice, the 6 will occur one time, right? — spirit-salamander

Wrong. A probability is not a surety of what will happen. A probability of 1/6 of the occurrence of the six doesn't ensure that the six will occur within 6 rolls. And the rest of your argument is arbitrary confusion.

The probability of 6 rolls all turning up the six is 1/(6^6).

The conditional probability (from having 5 rolls all turn up the six) of all six rolls turning up the six is 1/6, which is also simply the probability of a roll turning up the six.

And empirically, if the gambler's fallacy weren't a fallacy, then in Las Vegas there would be a whole bunch of fabulously rich amateur dice players and the casinos all bankrupt instead of all the losers who leave town begging for spare change for a bus ticket and the casinos profiting by billions of dollars.

lack of knowledge is innocence — Metaphysician Undercover

If lack of knowledge is innocence, then you are a saint.

we were talking about a count, which is a measure of quantity, not an order — Metaphysician Undercover

In this context, there are two senses of 'count':

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five."

A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)). Different orderings may determine the same cardinality, so different counts(1) whose result is the same count(2) may imply different orderings .

the number is excluded by the infinite order which must occur prior to it — Metaphysician Undercover

R as defined below is a well ordering of the set of natural numbers:

R = {<n m> | (~n=3 & m=3) v (n<m & ~n=3 )}

And let 'w' stand for the set of natural numbesrs. w+1 is the ordinal of <w R>.

maybe after you explain an infinite number of times, I'll get it — Metaphysician Undercover

I doubt it.

The point is that to describe a count as a tuple is not a correct description of a count. You just don't get it. — Metaphysician Undercover

We don't describe a count as a tuple. You don't even know what it is that you don't get.

the OP does not make much sense — EricH

The poster refers to 'classical logic', but doesn't know what classical logic is.

the SEP article on Classical Logic Is this your understanding of the term classical logic? — EricH

I only glanced through that article just now, but it looks good to me. I have found that Stanford Encyclopedia of Philosophy articles on logic are excellent and beautiful to read.

Logic/math statements do not refer to any event (real or hypothetical) in the physical universe, but are only true or false depending on the rules within the particular mathematical/logical system framework being used. — EricH

That is not universally accepted by all philosophers of mathematics.

The term "classical logic" is a bit vague, — EricH

Classical logic is exactly formalized. It's not vague.

Perhaps there is a way to translate into classical logic syntax (it's beyond my capabilities) but I'm reasonably confident that even if the sentence could be formulated it would have a value of false — EricH

The subject is explicated by Tarski's Theorem. If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.

Put differently, the sentence "This sentence is false" does not express a coherent thought — EricH

'Express a coherent thought' is an informal notion. Without formalization, there is no effective procedure by which in general people may objectively and definitively determine whether or not something "expresses a coherent thought". Mathematical logic though deals with 'this sentence is false' in a formal way that is clear and objective.

even if the sentence could be formulated it would have a value of false — EricH

This bears repeating: If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.

now I am used to being “a voice crying in the wilderness”. — Trestone

To get out of the wilderness, you could you formulate your notions in a way that other people can follow them, step by step, from basic to more involved.

maybe unconsciously I want to be the only one
who understands Layer Logic, — Trestone

You're doing a great job to make sure that you are.

The tuple notation is defined in mathematics.

And, of course, for sequences of length at least 2, there are different permutations. That there are permutations does not affect the count, since the count is the greatest number in the range, which remains constant under permutation.

You don't know anything.

You just don't get it.

"Refer" [...] means that we must direct our attention toward whatever it is which is referred to, — Metaphysician Undercover

Then:

(1)

You might still use "2" to name the book — Metaphysician Undercover

doesn't belong here. We do not use '2' to name a book.

(2) It seems your 'refer' might be close to what I mean by 'to pair with' or, more everyday, 'to associate with'.

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. Literally. We say the numbers, one for each object as we count the objects. Mathematically. this is expressed as a function from the set of things counted to a set of numbers:

{<'Portnoy's Complaint' 1> <'War And Peace' 2>}

That's a mathematical rendering of picking up 'Portnoy's Complaint' and saying '1', then picking up 'War And Peace' and saying '2', and if those are the only books, then saying 'The count is 2'.

Which pair is the true representation of the count? — Metaphysician Undercover

It is the very point that you can count more than one way.

You can count 'War And Peace as the first, then 'Portnoy's Complaint' as the second. Or you can count 'Portnoy's Complaint' as the first, then 'War And Peace' as the second. In either case, both counts show that there's a first and second, thus there are two.

Everybody knows that but you.

You seem to expect others to understand your notations without your having explained them from the bottom up. It borders on solipsism.

Do you understand the notion of either using commonly known notation or explaining our own personal notation starting at its most simple?

You look at your bookshelf, number "Portnoy's Complaint" as 2, and bring it in to me, telling me you have two books in your hand, because "Portnoy's Complaint" is identified as two books. — Metaphysician Undercover

I don't do that.

You present as so confused that I wonder whether you are posting as some kind of stunt or dumb cluck character.

We can have a count of 2 [electrons] without establishing the principles required to distinguish one from the other — Metaphysician Undercover

we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential — Metaphysician Undercover

We may infer, by whatever means, that there are a certain number of electrons or volts. That doesn't contradict that when we see discrete objects then we may count them.

Scientific measurement may have its special considerations. Or even everyday situations such as one glass of water having 8 ounces. But the question here is simple counting. How we use the concept of counting is a matter of practical approach, such as putting the water in a beaker with lines and counting the lines in the beaker to the point the water level ends or whatever. Whatever difficulties there may be conceptually with that, they don't negate the more basic notion of counting by bijection.

You might still use "2" to name the book — Metaphysician Undercover

You're doing it again! We do not use '2' to name a book. '2' does not denote a book.

neither Portnoy's complaint nor "War and Peace" need to be paired with either 1 or 2, for there to be a valid count of 2 — Metaphysician Undercover

We can switch them so that we have:

{<'Portnoy's Complaint' 1> <'War And Peace' 2>}

But the greatest number in the range is still 2.

if we remove "War and Peace", there is no longer two books, and the pairing is invalidated — Metaphysician Undercover

Bijections are not 'validated' or 'invalidated'.

The bijections

{<'War And Peace' 1> <'Portnoy's Complaint' 2>}

{<'Portnoy's Complaint' 1>}

of course are different, but nothing is "invalidated". Saying the pairings are "invalidated" is not even sensical.

the number 5 loses its meaning if it does not refer to five of something counted, — Metaphysician Undercover

The numeral '5' has meaning. The number 5 is not the numeral '5'.

5 is the count of a set of five books. 5 is the count of a set of five apples. 5 is the count of the set with two books, one apple, one house, and one person.

The fact that 5 is a count doesn't contradict that 5 also is a number no matter what it happens to count.

5 is the successor of 4. 4 is the successor of 3. 3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0.

No matter what the numbers count, they exist by virtue of successorship or by being 0.

we cannot dispense with the fact that "1" must refer to the object being counted, a book
— Metaphysician Undercover — TonesInDeepFreeze

I'm not saying a number is a book — Metaphysician Undercover

If I'm not mistaken, in another thread, you were using the word 'refer' in the sense of 'denote'. So if not 'denote' what exactly do you mean by 'refer' in this thread?

~0=1 Trestone: true in layer math — Trestone

What is the proof in layer math?

~Ex (x is a natural number & x>x) Trestone: true in layer math — Trestone

What is the proof in layer math?

In general, without stating axioms and a proof system, how do know what is true in layer math? Is there a layer math phone hotline you call and they tell you what's true or false?

If "2" denotes "Portnoy's Complaint" — Metaphysician Undercover

It doesn't.

You are inference impaired.

Read the whole introduction. — frank

I read it. She doesn't say anything on page 4 about set theory not being proven. At an earlier point, she does mention that the continuum hypothesis is not provable from the axioms. If that's what you have in mind, then it is not even close to saying that "set theory is unproven". You seem not to understand what set theory is when you say "set theory is unproven".

I still think English isn't your first language. You're doing great, though. — frank

Your sophomoric sarcasm is misplaced.

's basically what I said when you first took exception:
— TonesInDeepFreeze — frank

I didn't write that. I wrote:

It's basically what I said when you first took exception: — TonesInDeepFreeze

I don't know what snagging you think there is. You made an unnecessary rally about the matter even after I gave you ample clarification.

Perhaps you went off course when you overlooked that I included the word 'MIGHT' [emphasis added here] just as she did.

And I still don't know what you mean by

It might go the same way it came — frank

or what you mean by

Read page 4 where she explains the problems that arise from the fact that set theory is unproven: — frank

since she doesn't say anything about "set theory is unproven" or even what one would mean by "set theory is unproven".

I really thought English wasn't your first language. — frank

Then you're ridiculous.

That sounded better. — frank

It's basically what I said when you first took exception:

by saying "might" she's making clear that at that point she is not herself saying that talk about infinite numbers is not to be taken seriously. — TonesInDeepFreeze

Since you stooped to a cheap shot with "Is English not your first language?", I'll do you the favor of correcting your English:

it's truth — frank

it should be 'its' there.

The whole book is good though (for lay people like me.) — frank

The period should be after the right parenthesis.

It's a fair paraphrase. If you misunderstood me, then I would have been better just to quote her [by 'the whole situation' she means the difficulties in deciding whether Cantor's infinities are discovered or invented]:

"Finally, the whole situation might be interpreted as evidence that talk of infinite numbers is not really to be taken seriously"

Then she goes on to mention how some people argue for the position that talk of infinite numbers is not to be taken seriously. Clearly, she is presenting that argument not necessarily as her own position but rather to explain the views of those who do ascribe to the argument.

As is typical in such writing, she temporarily argues on behalf of others in order to explain their views but later goes on to examine those views from outside.

I understand it well.

Read page 4 where she explains the problems that arise from the fact that set theory is unproven — frank

I don't see anything there about set theory being unproven. I don't know what sense of 'unproven' you have in mind. The theorems of set theory are provable from the axioms of set theory, while of course the axioms are not proven except in the trivial sense that an axiom on a line alone is a derivation. However, neither the continuum hypothesis nor its negation are provable from ZFC (if ZFC is consistent, which is a "background" assumption in discussion of independence), which she mentions earlier, so maybe that's what you have in mind.

At that point in the book, she is entertaining the idea that talk about infinite sets is not to be taken seriously. Of course, the book is a presentation of various points of view about infinity, so by saying "might" she's making clear that at that point she is not herself saying that talk about infinite numbers is not to be taken seriously.

she's laying out an existing viewpoint. It's not hers. — frank

I didn't say that it is her view that talk about infinite sets is not to be taken seriously. I said that she mentions that the difficulties '"MIGHT" be taken as evidence that talk about infinite numbers is not to be taken seriously.

Page 3, last paragraph.

Yes. And other than that, in the Introduction, I don't know what you mean by "It might go the same way it came".

Perhaps you have in mind her idea that difficulties in the question of whether infinite numbers are discovered or invented might be taken as evidence that talk about infinite numbers is not to be taken seriously.