The notion of an actual infinite makes zero sense if, as per my assumption, actual means what it seems to mean to wit, completed in one sense or another for it flies against the definition of infinity as being necessarily that which can't be completed. — TheMadFool

Yes. It's a contradiction.

Maths, set theoretical infinities, kind courtesy of Georg Cantor, is an altogther different story as maths is essentially an axiomatic system, anything goes so long as you don't contradict yourself within one. — TheMadFool

It might go the same way it came (per Mary Tiles).

Where does one find a definition of 'actual' that includes 'completed'? — TonesInDeepFreeze

It might go the same way it came (per Mary Tiles). — frank

What do you mean by that? And are you referring to her book 'The Philosophy Of Set Theory'? If so, what in particular do you have in mind from that book?

What do you mean by that? And are you referring to her book 'The Philosophy Of Set Theory'? If so, what in particular do you have in mind from that book? — TonesInDeepFreeze

In the introduction she maps out the intellectual landscape pretty straightforwardly. The whole book is good though (for lay people like me.)

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.

Did she say that?

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

Where did she say that?

Page 3, last paragraph.

Page 3, last paragraph. — TonesInDeepFreeze

If you read that again, you'll see she's laying out an existing viewpoint. It's not hers.

Read page 4 where she explains the problems that arise from the fact that set theory is unproven: problems with how we might assess it's truth and problems with set theory's relationship to the real world.

She isn't advocating the downfall of set theory. She's explaining its unresolved and pretty significant philosophical problems.

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.

Oh. You called it "her idea", so I misunderstood.

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.

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. — TonesInDeepFreeze

You're misunderstanding that passage. Is English not your first language?

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.

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.

That sounded better.

So she lays out the conflict between finitists and set theory advocates and talks about how we might resolve the conflict.

It seems that the only route available is an indirect one: explaining the costs of going with finitism and understanding the value set theory has to math. The rest of the book fleshes all this out.

With this kind of approach, it's pretty clear that set theory is different from the kind of math most people learn. For instance arithmetic goes hand in hand with clear intuitions. There's no alternative viewpoint. There are no suspicions that arithmetic is contrived to provide some value internal to math itself.

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

That wasn't a cheap shot. I really thought English wasn't your first language.

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

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

Then you're ridiculous.

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

Wow. You really got snagged on that one.

Then you're ridiculous. — TonesInDeepFreeze

Certified, buddy.

'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".

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

Read the whole introduction. I still think English isn't your first language. You're doing great, though.

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.

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


Your sophomoric sarcasm is misplaced. — TonesInDeepFreeze

Just a tidbit: completely proper grammar gives you away as a non-native. Native speakers are usually slack. That's how it is with Spanish, anyway.

You are inference impaired.

You are inference impaired. — TonesInDeepFreeze

I don't think I have that certification. What agency takes care of that?

What's interesting about this is that whereas it is quite easy to see how mathematics (at its extremes) makes no sense, everything else knowable is EXACTLY the same. It's just more difficult to see. — synthesis

Oh I see I missed your point the other day and this is a good point. Yes even hard science is ultimately nonsense. There are no quarks. If you drill it down far enough you get to something that can't be quite right. Is that what you meant?

And social sciences like history are like that too. Something incredibly complex happens out of the interactions of thousands of people, then we label it "The Peloponnesian war" or "The industrial revolution," but these are just abstractions that summarize so many individual actions and events that in the end the abstraction must be a lie.

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

A number (we're talking about natural numbers in this context) is a count in sense (2). That doesn't preclude that a number is a mathematical object. — TonesInDeepFreeze

That's right, it doesn't preclude that the number is a mathematical object. But the point is that your definition (2) stipulates "the result of counting". So correct use of "5" is dependent on the count of the books, that there are five books. Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. Anytime we use "5" regardless of whether you think it refers to a mathematical object or not, it necessarily refers to five distinct units, or else you are using it incorrectly.

We better dispense with that notion. It's nuts. A number is not a book. — TonesInDeepFreeze

I'm not saying a number is a book, that's nonsense. But when we use "5" it is necessary that there are five distinct units indicated in that usage or else you are using "5" in an unacceptable way. Do you agree?

So the numeral does not denote a book, but rather it denotes the number that is paired to the book in the bijection (or, in everyday terms, in the pairing off procedure we call 'counting'). — TonesInDeepFreeze

Strictly speaking this (bijection in the way you describe it) is not a valid count. Suppose we say that there are two books. "War and Peace" is numbered as 1, and "Portnoy's Complaint" is numbered as 2. The relation between "Portnoy's Complaint" and the number 2 is not a simple pairing. This is evident from the fact that if we remove "War and Peace", there is no longer two books, and the pairing is invalidated. You might still use "2" to name the book, but it is not a valid count of two, because there is only one book.

So we cannot say that "Portnoy's Complaint" is paired with 2. That is a false representation because it does not include the necessary requirement of another book. "Portnoy's Complaint" can only be paired with 2 in a valid count, if there is another book paired with one. Furthermore, 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. Do you recognize this point? There is no need for a pairing to have a valid count. We can have two objects, and say that there are two, without naming either as one or two, they are simply two.

This latter point is something which is very important to understand, especially when we count things like electrons which are difficult to distinguish from one another. We can have a count of 2 without establishing the principles required to distinguish one from the other. We can say that there are two electrons in the same orbit, without the need of distinguishing one from the other. We have principles which say they are distinguishable, but we need not distinguish them. Likewise, we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential, as 1,2,3, etc..

So it is very clear that your method of representing "a count", as pairing a number with a unit (bijection) is a totally inadequate representation of what a count really is.

We don't say "''1' denotes 'War And Peace' and '2' denotes 'War And Peace' together with 'Portnoy's Complaint'". That's crazy. — TonesInDeepFreeze

You think it's crazy, but it's what's required to have a valid count. If "2" denotes "Portnoy's Complaint", unconditionally, and you have no other books, then obviously your count of 2 books is invalid. If you deny this requirement them you allow for invalid counts. 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. That's what's really crazy.