:up:

That means you understand why it's not such a good idea to post disinformation that set theory claims that a (proper) part is equal to the whole?

And maybe, you'll say who you thinks says that set theory does make that claim?

But you're a fan of physics so you should appreciate that in QM the observer plays an active role converting the potential to the actual (by means of making measurements to collapse the wave function). — keystone

I'm a fan of semiotics – the modelling relation. So that is a formal theory of how observers and their realities relate by acts of measurement.

I'm working one step up. I'm speaking for those like Peirce, Rosen, Pattee, etc, who are modelling the modelling relation in a rigourous fashion. And yes that does then lead to a better metaphysical understanding of both classical and quantum physics.

A mathematical platonist would have to say infinite. I would go in a different direction. I would ask 'where?' How many numbers are 'where'? In other words, in what computer/mind are you talking about. You have to be specific about where is because there is no Platonic realm. — keystone

I am saying much the same thing. But the question is not where the numbers need to be represented or stored. It is how many decimal places do you really need for the task in hand?

What this means is that the whole business of counting becomes self-limiting. The aim of "good mathematics" is instead to represent the fewest digits you can reasonably get away with. An infinity might be available, but I would rather save time and energy for other things by only having to remember just one.

This is where we get down to binary bits of logic. The only numbers needed are 0 or 1. Or indeed, the tape head of a Turing machine that can either make or erase a mark.

But it takes the semiotic view to create this reciprocally self-limiting Nirvana – where infinite information is available, yet you can boil it all down to a simple yes/no response.

Is it a 0 or a 1? That's all I need to know to be completely certain rather than maximally uncertain. And a world that is boiled down to yes/no certainty demands hardly any time or energy to live in it.

This is what maths looks like when it does involve an intelligent observer with some actual purpose. A big enough range of possibilities to cover all eventualities. But then the complementary operation that narrows the field to a single actuality of complete certainty or counterfactual definiteness.

I take "approaches" to be a potentially infinite process. — keystone

That just takes the conversation back to where we were before. One can have whatever concept of limits one wants to have, including conceiving in terms of potential infinity. Indeed, there are systems that do (and I little doubt succeed) in axiomatizing large amount of analysis without infinite sets.

And lots of people aren't concerned with axiomatization, so they don't even care whether or how analysis is axiomatized, as they at the same time prefer to conceive in terms of potential infinity rather than there being infinite sets. Not at issue.

But, to the extent that one is interested in axoimatization, one would want to go the next step, which is to ask, okay, what are the axioms, or at least, what initial ideas are there for what the axioms might be?

If you say, that's not your concern, then fine. Then we just have different roads we want to travel. But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not.

Ok!

Okie dokie cranky spanky.

disinformation — TonesInDeepFreeze

Most interesting. — Ms. Marple

While I do agree that mathematicians have achieved some kinda broad consensus on the role of $\infty$ in mathematics, probably this is your area of expertise, the existence of finitism suggests to me that there's trouble in (Cantor's) paradise!

the existence of finitism suggests to me that there's trouble in (Cantor's) paradise! — Agent Smith

There are critics of X, therefore there is something very wrong with X.

That is a risibly stupid argument.

There are critics of X, therefore there is something very wrong with X.

That is a risibly stupid argument. — TonesInDeepFreeze

Criticisms by mathematicians (finitism is a mathematical movement), to my knowledge, aren't usually baseless.

disinformation
— TonesInDeepFreeze — Agent Smith

Then you write a post having nothing to do with your quote of me. It still stands that it is disinformation to say that set theory claims that a (proper) subset of the whole is equal to the whole.

The argument for the claim I made is a simple one and perhaps it doesn't meet the standards of rigor required in mathematics.

There's nothing in the world that doesn't have detractors. So, by your logic, everything is wrong.*

Finitism has detractors, so by your logic, there's something very wrong with finitism.

* Or, to put it not so sweepingly, let's unpack what you've said.

(1) As I've pointed out to you probably half a dozen times already, finitism has many forms. Some finitists work in infinitistic set theory.

(2) Many finitists are critics of infinitistic set theory. But your own criticisms are ignorant, self-malinformed, and disinformation. You don't like the concept of infinite sets and want to see a mathematics without them. That is fine. But you grab any chance to indulge your confirmation bias about the subject.

In this case, you don't even appeal to specifics but instead ludicrously reason that since there are critics then we can pretty much bet that they're right. That is so remarkably irrational. It's in the same league of irrationality as people who say, "The conclusion that climate change is anthropogenic is wrong, which I know because there are scientists who say it's wrong."

No, to claim that the conclusion is wrong requires actually comparing the work of the dissenting scientists with the work of the preponderance of other scientists.

To claim that set theory is wrong requires comparing the arguments of the detractors of set theory with the arguments of the mathematicians and philosophers in favor of and in defense of set theory.

Just grabbing random out of context quotes against set theory, one-liner tidbits, and polemical irrelevancies to argue against set theory is like viewers of Fox News who base their claims about politics on whatever chyrons happen to cross the screen, whatever infantile propaganda memes are splashed and whatever disassociated falseoids happen to spill from the mouths of the on-screen anti-pundits.

it doesn't meet the standards of rigor required in mathematics. — Agent Smith

It's not so much that you don't meet a standard of rigor, it's that you lie about the subject.

Ok! Google should help you find the relevant pages! Bonam fortunam.

It's not so much that you don't meet a standard of rigor, it's that you lie about the subject. — TonesInDeepFreeze

:rofl:

I can't speak to the standard axiomatization of analysis, but the informal definitions that us engineers were taught didn't use sets — keystone

If it's any consolation I was a math major at two big state universities in the 1950s and I can't recall of ever studying any aspect of foundations beyond skimpy material on rationals and irrationals and the continuity of the real line. As a freshman at Georgia Tech I was placed in an experimental course in introductory calculus that began with epsilons and deltas - very confusing at first. The engineers had it much easier, as I learned when I moved back into the standard curriculum. But when I started grad school at another university in 1962 one of the first required courses was an introduction to foundations using Halmos' Naive Set Theory and the Peano Axioms. It was quite illuminating.

Ok! Google should help you find the relevant pages! Bonam fortunam. — Agent Smith

What? You're the one spreading disinformation, notwithstanding the little touch you give with Latin phrases.

You say that set theory claims that a (proper) part can be equal to the whole. I explained to you twice exactly the way you are incorrect. My going to Google or not doesn't affect that you're spreading disinformation.

It's not so much that you don't meet a standard of rigor, it's that you lie about the subject.
— TonesInDeepFreeze

:rofl: — Agent Smith

You'd be right if that emoticon meant 'QED'.

:smile: Good day señor!

Gaudeo te relinquere, domine.

There, see, I took you up on your suggestion to visit Google.

How is that substantively different from Thompson's lamp?
I already responded regarding Thompson's lamp.
I don't know a theorem of set theory that is rendered as "infinite processes can be completed".
Set theory doesn't axiomatize thought experiments. — TonesInDeepFreeze

I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to completion. However, I think set theory can be reframed to correspond to potentially infinite algorithms instead of actually infinite sets. After all, we never directly work with the infinite sets themselves, but instead the finite strings of characters that describe them.

Thought experiments are beneficial because they make it clear what we're talking about - with set theory (as it is framed today) we're talking about actually infinite objects, not potentially infinite processes. And I'm not convinced by your response to Thompson's lamp because your answer lies outside of the thought experiment where it's unclear to me whether your resolution requires the completion of an infinite process. In Hilbert's hotel universe, infinite processes can be completed so there must be a definite final state of the lamp. If you can't provide the state of the lamp then it's worth questioning whether infinite processes can truly be completed in that universe. And if they can't even be completed in an imaginary universe, why would we think they can be completed in reality?

But that doesn't prove that there does not exist a set whose members are all and only the natural numbers or that there does not exist an infinite set. — TonesInDeepFreeze

A finite being cannot exhibit or work with an infinite set directly. To do so requires an infinite being. Since one cannot disprove the existence of an infinite being (e.g. God) one cannot disprove the existence of infinite sets. But does the burden of proof lie on the atheist? Paradoxes are a good option because they demonstrate that actually infinite universes (in which infinite sets can exist) harbor contradictions, such as in Hilbert's Hotel Universe where infinite processes can and cannot be completed.

Extensionally, Hilbert's Hotel refers to the trivial possibility of indefinitely expanding a finite hotel in such a fashion that guests are reassigned to new rooms as new guests are added. Unfortunately, ZFC cannot distinguish between a hotel that isn't finite purely because it is growing without bound, from a mythical hotel with a countably infinite subset, which as you point out, is an extensionally meaningless assertion, and is partly the fault of the axiom of choice that ZFC assumes.

That "A Hilbert Hotel has a countably infinite subject" refers to a sentence of ZFC, and not an actual hotel.

Hilbert's Hotel refers to the trivial possibility of indefinitely expanding a finite hotel — sime

The hotel is not finite. It has infinitely many rooms.

ZFC cannot distinguish between a hotel that isn't finite purely because it is growing without bound, from a mythical hotel with a countably infinite subset — sime

ZFC doesn't distinguish among hotels, real or mythical.

And the point is not that there is a countably infinite subset. Every countably infinite set has a countably infinite subset.

the fault of the axiom of choice — sime

I don't see what the axiom of choice has to do with it. The rooms are enumerated by room numbers. Choice is not invoked.

"A Hilbert Hotel has a countably infinite subject" refers to a sentence of ZFC, and not an actual hotel. — sime

Hilbert's Hotel is an imaginary analogy to ('S' for the set of positive natural numbers):

For any natural number k>0, S is 1-1 with S\{1 2 ... k}.

For k = 1, as a new guest arrives, we move each already staying guest to the room above.

For k>1, as k number of new guests arrive all at once, we move the already staying guests up more than one room, as we move them up k number of floors.

/

For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle.

When you throw a dart at a dartboard, you don't hit a point, you hit an area. Any discretization of a dartboard into areas produces a finite number of areas each with a finite probability, all summing to a probability of 100%. What's wrong with this view? — keystone

Nothing is wrong with this view except that it misses the point of the paradox, which isn't related to a literal physical dart (we have no idea ifphysical space is discrete or not; this is a debated metaphysical topic I won't get into) rather the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).

Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?"

In any case, there are two mathematically respectable solutions to this 'paradox'. One is the philosophical thesis of saying that zero-probability is not the same as modal impossibility, which is so far the most widely accepted solution. The other solution is to introduce hyperreal numbers, particularly nilpotent infinitesimals, such that each contestant has probability ε but reiterative summation does not eventually yield anything over 100%, primarily because while ε isn't 0, ε+ε can still equal 0. This approach is not as widespread.

The hotel is not finite. It has infinitely many rooms. — TonesInDeepFreeze

A perpetually growing hotel that always has a finite number of rooms is still an infinite set, because there isn't a bijection between any finite set and the number of rooms in the hotel. But such a hotel isn't describable in ZF if the axiom of choice is assumed, because it forces Dedekind-infiniteness upon every infinite set.

A perpetually growing hotel — sime

As I recall, it's not a perpetually growing hotel. Rather, it's a hotel with denumerably many rooms and denumerably many guests, one to each room.

The "paradox" is not about potential infinity, but rather about the set of natural numbers (or any denumerable set along with a given enumeration of it).

But such a hotel isn't describable in ZF if the axiom of choice is assumed, because it forces Dedekind-infiniteness upon every infinite set. — sime

You have it backwards. That the set of rooms is Dedekind infinite is what makes the hotel "paradoxical".

Moreover, we don't need any choice axiom to prove that the set of natural numbers is Dedekind infinite.

The "paradox":

Let there be a hotel with denumerably many rooms with room numbers 1, 2, 3 ...

Suppose there are denumerably many guests in rooms and that each room has a guest.

Suppose a new guest arrives. Then the hotel manager moves the already staying guests this way: If a guest is in room j, then that guest is moved to room j+1. And the new guest is put in room 1.

Or suppose k (k a natural number such that k>1) number of new guests arrive. Then, for all guests, if the guest has been in room j, then the guest moves to room j+k. And the k new guests are put in rooms 1 through k.

/

Indeed it is the failure of the pigenonhole principle for infinite sets (i.e. that infinite sets are Dedekind infinite) that allows the "paradox".

As I recall, it's not a perpetually growing hotel. Rather, it' a hotel with denumerably many rooms and rooms and denumerably many guests, one to each room. — TonesInDeepFreeze

That's right. But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. The latter refers not to a hotel, but to a piece of syntax representing an inductive definition of the natural numbers.

The paradox is due to conflating intension with extension. Keystone is right to raise objection.

Since one cannot disprove the existence of an infinite being (e.g. God) one cannot disprove the existence of infinite sets. — keystone

Your analogy betwen mathematics and theology is not apt.

One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.

Anyway, I never asked you to disprove anything at all.

infinite universes (in which infinite sets can exist) harbor contradictions — keystone

A contradiction is a certain kind of sentence. But, of course, there is no world in which a contradiction is true.*

* The context here is ordinary logic.

And there is no contradiction in set theory.*

* As far as we know.

We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately.

Now, if you wish to have a mathematical theory, adequate for science, that does have a domain of discourse of physical particles and physical objects, then no one is stopping you from saying what that mathematical theory is, or might be, or some idea of it. Saying, [paraphrase] "We'll keep set theory except infinite sets and use potential infinity instead" is suggestive of an idea, but not much more.

where infinite processes can and cannot be completed. — keystone

I know of no theorem of set theory that there are infinite processes that both can and not be completed.

I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to completion. — keystone

You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle.

However, I think set theory can be reframed to correspond to potentially infinite algorithms instead of actually infinite — keystone

Same as above.

I'm not convinced by your response to Thompson's lamp because your answer lies outside of the thought experiment — keystone

The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. One sets up a thought experiment, then suggests an anology with set theory, as that analogy however is only informal and imaginary. Then I immediately concede that set theory doesn't explain informal, imaginary analogies to it. The terms of the thought experiment are not set theory.

Set theory doesn't have lamps that turn off and on infinitely but such that there's a final state in which the lamp is either off or on.

But set theory does have infinite sums if there is convergence. So, set theory does not say there is such a "final state" for a non-converging sequence of 0s and 1s. Set theory doesn't have a contradiction that there is a final state for such a sequence. That's something good about set theory.

But one can say, what about the fact that set theory has the finite ordinals, but then the least infinite ordinal that comes after all the finite ordinals? Yes, but no one says that there is "process" by which we go trough all the finite ordinals and then arrive at the least infinite ordinal.

why would we think they can be completed in reality? — keystone

We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary.