Philosophy Now Forum

wtf wrote: ↑Thu Mar 31, 2022 1:35 am

Eodnhoj7 wrote: ↑Wed Mar 30, 2022 11:50 pm
The sequence 0, 1, 2, 3, 4, 5, ... is incomplete. In reality it is -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... thus the center is zero; the center is empty.

Since the sequence is infinite in both directions, how do you know the center isn't, say, 6? After all there's a bijection between the set of integers less than 6, and the set of integers greater than 6.

Unless one infinity is greater than another then all numbers are center points. However if this is an infinity +1 or +2 then the center is fixed.

wtf wrote: ↑Thu Mar 31, 2022 5:45 am No, that's not right at all.

I'm sure you know that a set, without any additional structure, has no inherent order. The set {banana, orange, apple} is the same as the set {orange, apple, banana}. A set is characterized only by its elements.

Now apple, orange, and banana are not mere "placeholders." They are elements of the set. Just because the order doesn't matter in the set, doesn't mean the elements can't be distinguished.

If you're at the grocery store, it doesn't matter if they put the apple, orange and banana in the grocery bag in that order, or in some other order. Either way, the set of groceries is exactly the same because it always has exactly the same contents.

But the elements themselves, the apple, orange, and banana, are still distinct elements different from one another. Surely you see that, right?

This is a terrible. Absolutely terrible analogy! You are mixing up two things.

1. The topology of the set being discrete. And I am not using a formal definition here - am simply pointing out that you can pick out elements of the set independently from one another.
2. The inherent properties of the elements of the set being different.

Given the set {apple, orange, banana}.

Fruits have inherent properties such as taste, size, smell, color, texture, shape etc etc.
The properties of banans, apples and oranges are what makes them different from one another.
It's precisely those properties which allow you to tell whether what you've taken out of the bag is an orange, banana or an apple.

Now take the set {1,2,3}. Numbers don't have any inherent properties. What makes 1 different from 2, different from 3? How are you distinguishing the elements of the set while they are still in the set? Lacking any inherent properties AND lacking any ordering they are all the same with respect to ALL of their properties. And identity with respect to ALL properties is one of the many definitions of equality/equivalence.

wtf wrote: ↑Thu Mar 31, 2022 5:45 am Likewise, if we merely called an apple a banana and a banana an apple, it also wouldn't change the contents of the bag. And there would still be one of each fruit, they'd just have different names. And they'd still all be distinct from one another no matter what name you call them and no matter what order you place them in the grocery bag.

But of course the names of things matter! By virtue of insisting that objects in the set are uniquely identifiable you are literally giving structure to the set! Certainly more structure than just "a bag of stuff".

The requirement of unique identifiability of elements is literally what distinguishes {1,2,3,4,5,...} from {1,1,1,1,1,...}

It's obvious that the first set contains infinitely many uniquely identifiable elements, while the 2nd set contains infinitely many identical elements. And any logicial should be able to infer the following implications:

FORALL x, y members of {1,2,3,4,5,...} x <> y holds.
FORALL x,y members of {1,1,1,1,1,...} x = y holds.

Which, of course, is trivially true to any clasical logician: FORALL x, y members of ANY set S -> equal(x,y) OR not(equal(x,y)) by Law of Excluded Middle.

Now if you want to object to that (like any reasonable constructivist would)... so be it - we welcome you with open arms. But set theory founded upon classical logic is just nonsense.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm This is a terrible. Absolutely terrible analogy! You are mixing up two things.

1. The topology of the set being discrete. And I am not using a formal definition here - am simply pointing out that you can pick out elements of the set independently from one another.
2. The inherent properties of the elements of the set being different.

If you're trolling, it's pretty weak. If you're just ignorant of set theory, I'll do my best to educate you. This has nothing to do with topology. The elements of a set are distinct by definition.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm Given the set {apple, orange, banana}.

Fruits have inherent properties such as taste, size, smell, color, texture, shape etc etc.
The properties of banans, apples and oranges are what makes them different from one another.
It's precisely those properties which allow you to tell whether what you've taken out of the bag is an orange, banana or an apple.

Now take the set {1,2,3}. Numbers don't have any inherent properties. What makes 1 different from 2, different from 3? How are you distinguishing the elements of the set while they are still in the set?

In set theory, 1, 2, and 3 are distinct sets. 0 = ∅, 1 = {0}, 2 = {0,1}, and 3 = {0, 1, 2}. This is the von Neumann definition of the counting numbers. As you can see, these are all distinct sets.

https://en.wikipedia.org/wiki/Set-theor ... al_numbers

There are other ways of representing the counting numbers as sets, but in every such method, 1, 2, and 3 are different sets. That's the whole point.

The reason von Neumann's idea won out is because it extends naturally to the transfinite ordinals. The other methods don't.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm Lacking any inherent properties AND lacking any ordering they are all the same with respect to ALL of their properties. And identity with respect to ALL properties is one of the many definitions of equality/equivalence.

All that's lacking is your knowledge of basic set theory. 1, 2, and 3 are distinct sets. Now that I've explained to you what 1, 2, and 3 are in set theory, you are educated and a little bit less ignorant than you were before. You're welcome.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm The requirement of unique identifiability of elements is literally what distinguishes {1,2,3,4,5,...} from {1,1,1,1,1,...}

It's obvious that the first set contains infinitely many uniquely identifiable elements, while the 2nd set contains infinitely many identical elements. And any logicial should be able to infer the following implications:

Sets have no duplicate elements. The set {1, 1, 1, ...} and the set {1} are exactly the same set. Its cardinality is 1.

Of course there are multisets, but even then, the multiplicity is restricted to being a finite natural number. See

https://en.wikipedia.org/wiki/Multiset#Definition

I suppose you could extend the definition to allow transfinite cardinalities, but they would still not be sets. Sets have no repetitions.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm FORALL x, y members of {1,2,3,4,5,...} x <> y holds.
FORALL x,y members of {1,1,1,1,1,...} x = y holds.

This is perfectly true (and entirely trivial) by virtue of the axiom of extensionality, which says that a set is entirely characterized by its elements.

https://en.wikipedia.org/wiki/Axiom_of_extensionality

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm Which, of course, is trivially true to any clasical logician: FORALL x, y members of ANY set S -> equal(x,y) OR not(equal(x,y)) by Law of Excluded Middle.

True and trivially so. The fact that you are ignorant of a particular subject doesn't make your observations clever. And now that I've pointed you to the definition of 1, 2, and 3; and the definition of a multiset; and the axiom of extensionality; you are thereby that much less ignorant than you were before you read this post.

Skepdick wrote: ↑Mon Apr 25, 2022 2:12 pm Now if you want to object to that (like any reasonable constructivist would)... so be it - we welcome you with open arms. But set theory founded upon classical logic is just nonsense.

Oh, ok. You think set theory is nonsense. Since you've demonstrated your total ignorance of the most fundamental principles of set theory, how can you say it's nonsense? At least have the intellectual honesty to learn something about the subject, rather than labeling as nonsense that which you simply don't understand.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm If you're trolling, it's pretty weak. If you're just ignorant of set theory, I'll do my best to educate you. This has nothing to do with topology. The elements of a set are distinct by definition.

This doesn't address anything of what I said! You are conflating two concepts into one: you are mixing up distinctness and identity!

Two copies of the same file are distinct ∧ they are identical.
The digits 1 and 1 are distinct ∧ they are identical.
The sets {} and {} are distinct ∧ they are identical.
The digits 1 and 2 are distinct ∧ they are NOT identical.
The sets {banana} and {banana, banana} are distinct ∧ they are NOT identical.

This shouldnt be difficult to grasp without the need for definitions - simply consult your intuition.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm In set theory, 1, 2, and 3 are distinct sets. 0 = ∅, 1 = {0}, 2 = {0,1}, and 3 = {0, 1, 2}. This is the von Neumann definition of the counting numbers. As you can see, these are all distinct sets.

https://en.wikipedia.org/wiki/Set-theor ... al_numbers

The reason von Neumann's idea won out is because it extends naturally to the transfinite ordinals. The other methods don't.

Again. This doesn't address my point about the implicit ordering of the sets!

If you are going to equate ∅ and 0 then what's the cardinality of ∅? 0? But 0 = ∅ ?!?
What's the cardinality of {0}? 1? But 1 = {0} ?!?

So the cardinality of any set is its identity morphism. That's viciously circular!

Otherwise, your insistence that the natural numbers don't have any ordering directly translates to there being no ordering to the cardinality of the elemens either.

If any ordering is order-isomorphic to the usual order then surely that applies to the cardinalities of sets also. and so we could have |{0}| > |{0,1,2} | > | {0,1}|

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm All that's lacking is your knowledge of basic set theory. 1, 2, and 3 are distinct sets. Now that I've explained to you what 1, 2, and 3 are in set theory, you are educated and a little bit less ignorant than you were before. You're welcome.

Maybe it's time to stop "educating" people on something you clearly don't understand.

1,2 and 3 are distinct sets. Nobody is disputing that. But they are not identical sets.
1, 1 and 1 are also distinct sets. And they are identical sets.

You continue conflating two different properties of sets: their distinctness and their identity.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm Sets have no duplicate elements.

Then don't interpret them as "duplicates"! They are just placeholders - they represent the distinct elements of any set. Whatever those elements may be - we are not interested in the elements or their properties because our set is not equipped with any relation. Right? Nowhere does it say that the elements of sets must be sets, or numbers or anything in particular. They could be urelements. Heck! You were the one who used a bag of fruit as analogy. So you are telling me I can't have a bag with more than one banana? That's just dumb. {banana} and {banana, banana} are different bags!

Also...I am not even going to jump into the intricacies of how 1 = 1 could even be meaningful thing to say in set theory without multisets. If the set of natural numbers contains no duplicate elements what is the arity of the equality relation? Surely it must be unary!

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm The set {1, 1, 1, ...} and the set {1} are exactly the same set. Its cardinality is 1.

That's literally and observably not true.

{1} has cardinality 1.
{1,1} has cardinality 2.
{1,1,1,...} has cardinality Aleph-Zero

Now, you are allowed to ignore this obvious difference and play pretend (and obviously that is what the set theorists are doing).

But any non-idiot can see that {1} and {1,1} are not "exactly the same set". All that is required of you is to use your eyes. In much the same way you can distinguish a bag with one banana from a bag with two bananas.

{banana} is not the same bag as {banana, banana}! If your theory defies such basic human intuition then junk the theory and keep the intuition.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm This is perfectly true (and entirely trivial) by virtue of the axiom of extensionality, which says that a set is entirely characterized by its elements.

https://en.wikipedia.org/wiki/Axiom_of_extensionality

We clearly understand that axiom very differently. The axiom states (quite plainly, in English) that sets having the same elements are the same set.

But there's this thing in logic... modus tollens and modus ponens.

IF P => Q.
Not Q => Not P.

If {1} and {1,1} have the same elements => {1} and {1,1} are the same set.
{1} and {1,1} are not the same set (they differ in cardinality) => {1} and {1,1} don't have the same elements.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm True and trivially so. The fact that you are ignorant of a particular subject doesn't make your observations clever. And now that I've pointed you to the definition of 1, 2, and 3; and the definition of a multiset; and the axiom of extensionality; you are thereby that much less ignorant than you were before you read this post.

I am not trying to be clever. I am trying to point out you are fairly stupid.

The triviality disappears when you abandon the axiom of excluded middle.

wtf wrote: ↑Wed Apr 27, 2022 9:29 pm Oh, ok. You think set theory is nonsense. Since you've demonstrated your total ignorance of the most fundamental principles of set theory, how can you say it's nonsense? At least have the intellectual honesty to learn something about the subject, rather than labeling as nonsense that which you simply don't understand.

I am trying to learn, but I am having a hard time overlooking the bullshit. It's really peculiar to me why set theorists would conflate the notions of distinctness and identity.

This clear disregard for qualitative semantic differences really assaults my intuitions and insults my sensibilities.

All in all, it is trivially obvious to me (despite the insistence of set theorists) that it's impossible to define arithmetic or number theory using set theory as a foundation!

Every damn operator in arithmetic is binary If set theory does not permit element multiplicity then you can't even express 1 + 1, 1-1, 1* 1; 1/1, 1^1. 1 mod 1 etc. You can't express of practical utility in such a stupid theory.

wtf wrote: ↑Thu Mar 31, 2022 4:49 am Really? Where are you going to store those infinitely many bits? Are you going to keep running down to Best Buy for more memory? Crucial has better prices.

Perhaps this is where we are constantly missing each other. In one paragraph you talk about abstract machines and in the next breath you keep appealing to buying physical memory. Turing machines have infinite memory, no? Or do you have to keep buying RAM for yours?

Almost as if you are flip-flopping between finittist and infinitist perspectives. Almost as if you keep flip-flopping between accepting and rejecting the realizability interpretation of constructivists.

But far more pecular to me is that you seem committed to one particular model of computation - Turing's model. What about more powerful models of computation. Such as Infinite time Turing machines?

Of course, I would understand how somebody who believes that there is no time in pure mathematics might get confused by the notion of Infinite Time Turing Machines.

wtf wrote: ↑Fri Feb 14, 2020 11:26 pm But as to pure math itself, there's no time in it. Time is a question of physics, not math.

Time is precisely what distinguishes Type I from Type II computations.

Trivially. If there is such thing as infinite precision real numbers then you can store ALL the digits of any particular infinite precision real number on a single Turing machine as an infinite stream of digits. No?

So I don't really get the whole hogwash of there being more real numbers than Turing machines. For every real number we can define an ITTM which exactly returns its digits. It's just Baire space. N^N. Same as R.

Now tell me something about arithmetic on real numbers. If you can.

Skepdick wrote: ↑Thu Apr 28, 2022 2:34 pm Turing machines have infinite memory, no?

No. A TM has unbounded memory, but always finite. If you need a million cells or a trillion cells or Graham's number of cells to do a particular computation, then the tape is that long. But there are always a finite number of cells.

A TM can not store a random (or noncomputable) sequence of bits.

Skepdick wrote: ↑Thu Apr 28, 2022 10:38 am
I am not trying to be clever. I am trying to point out you are fairly stupid.

We're done.

wtf wrote: ↑Thu Apr 28, 2022 10:27 pm No. A TM has unbounded memory, but always finite. If you need a million cells or a trillion cells or Graham's number of cells to do a particular computation, then the tape is that long. But there are always a finite number of cells.

Bullshit, dude.

...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares

— Turing 1948, p. 3

And so you can take any unbounded sequence representing an infinite precision real number and write it on the damn tape. Voila! A computable number.

3
.
1
4
1
5
9
2
6
5
3
5
8
9
7
9
3
2
3
8
4
6
2
6
4
3
3
8
3
2
7
9
5
0
2
8
8
4
1
9
7
...

wtf wrote: ↑Thu Apr 28, 2022 10:27 pm A TM can not store a random (or noncomputable) sequence of bits.

A TM is storing any sequences of bits you have put on its infinite tape! Is just an infinite tape - you can write anything you want on it!

The origin, randomness or computability of this sequence is none of the TM's concern, and as the Mathematician trying to reason about the situation at hand just treat it as the initial object of some category you are busy constructing.