What does it mean for something to be "logically possible"?

[beero1000]

Prove it.

Prove that you didn't answer the question? Prove that you don't know that 0 is an integer? Prove that 0 is an integer?

It's like you're not even trying...

That is not very convincing proof.

That's because it wasn't a proof at all. Do you know what a proof is?

 

[bilby]

You didn't actually answer the question. Do you know that 0 is an integer?

Prove it.

0 is defined as an integer. It's part of the definition of the word. It's included in the meaning.

Can you prove that cars are vehicles? Or that red is a colour? Or that a sparrow is a bird?

How can you possibly be so bad at this? Were you dropped on your head as a child?

 

[beero1000]

Prove it.

0 is defined as an integer. It's part of the definition of the word. It's included in the meaning.

Can you prove that cars are vehicles? Or that red is a colour? Or that a sparrow is a bird?

How can you possibly be so bad at this? Were you dropped on your head as a child?

Literally the first Peano axiom.

This is the best untermensche roll I've ever seen!

 

[untermensche]

Prove it.

0 is defined as an integer.

By who?

 

[untermensche]

0 is defined as an integer. It's part of the definition of the word. It's included in the meaning.

Can you prove that cars are vehicles? Or that red is a colour? Or that a sparrow is a bird?

How can you possibly be so bad at this? Were you dropped on your head as a child?

Literally the first Peano axiom.

This is the best untermensche roll I've ever seen!

I have no idea what you mean by a roll?

You mean the part where a bunch of clowns claim the negative integers END at zero and I try to educate them?

Or the clowns that think you can progress from nothingness?

That think it is possible to crawl out of a hole infinitely deep.

 

[bilby]

0 is defined as an integer.

By who?

Giuseppe Peano, et al.

It's axiomatic.

You can reject it, but you should understand that to do so is to reject all of arithmetic, and most of mathematics, and so you will be left needing to formalise your own arithmetic from scratch before you can proceed.

This seems like a rather serious undertaking, and not one for which I believe you to be equipped.

 

[untermensche]

Just for the record since I am on a roll.

Nobody will find a claim from me that zero is not an integer, though I have my doubts.

What has been claimed is zero is not a positive or negative integer.

It is not part of either of those infinite series.

- - - Updated - - -

By who?

Giuseppe Peano, et al.

It's axiomatic.

You can reject it, but you should understand that to do so is to reject all of arithmetic, and most of mathematics, and so you will be left needing to formalise your own arithmetic from scratch before you can proceed.

This seems like a rather serious undertaking, and not one for which I believe you to be equipped.

So tell me how he proved it.

Since you know he did it should take you mere seconds.

 

[bilby]

Literally the first Peano axiom.

This is the best untermensche roll I've ever seen!

I have no idea what you mean by a roll?

You mean the part where a bunch of clowns claim the negative integers END at zero and I try to educate them?

Or the clowns that think you can progress from nothingness?

That think it is possible to crawl out of a hole infinitely deep.

I believe he is referring to your attempt to extricate yourself from your hole by redoubling your digging efforts.

You can't dig your way out of a hole that's even finitely deep.

 

[bilby]

Just for the record since I am on a roll.

Nobody will find a claim from me that zero is not an integer, though I have my doubts.

What has been claimed is zero is not a positive or negative integer.

It is not part of either of those infinite series.

Don't wear yourself out moving those goalposts; the original claims are still there for all to see.

- - - Updated - - -

Giuseppe Peano, et al.

It's axiomatic.

You can reject it, but you should understand that to do so is to reject all of arithmetic, and most of mathematics, and so you will be left needing to formalise your own arithmetic from scratch before you can proceed.

This seems like a rather serious undertaking, and not one for which I believe you to be equipped.

So tell me how he proved it.

Since you know he did it should take you mere seconds.

Go and learn the meaning of the word axiomatic.

Axioms are not proven; they are the basis on which other truths can be proven.

 

[untermensche]

Don't wear yourself out moving those goalposts; the original claims are still there for all to see.

Show me.

You have a habit of saying things that are not true.

The funniest is the negative integers END at zero.

Even though zero is not a negative integer.

Go and learn the meaning of the word axiomatic.

Axioms are not proven; they are the basis on which other truths can be proven.

You're spewing nonsense you cannot back up.

You can in no way prove zero is an integer.

 

[bilby]

Show me.

beero1000 already did this once. Here it is again:

You're so infatuatiated with me you don't seem to be able to read a word I write.

I said zero is not part of the POSITIVE integers. It is not a part of that set. It is not the first or last positive integer.

Let's go on a journey to the recent past and see what happened, shall we?

You posted this:

Anything can be done by definition. You can say 0 is an integer by definition even though zero has features unlike any other integer.

I thought to myself "That's weird, it kinda sounds like you don't think 0 should be an integer". It might have just been weird phrasing though, so fine, let it slide. But then you posted this:

The integers are two separate series. The positive integers and the negatives. In both series the lowest integer is the START of the series. You cannot have a series without a start to it.

You missed zero in the integers again! So I responded:

This is your second post here where you imply that you don't think 0 is an integer. Weird.

Your responses were these strawmen implying that I said 0 is a positive integer.

Not strange at all to those capable of thinking.

Zero is not positive.

How one could think it is a positive integer is the strange thing.

I know. Very troubling when people are so lost they think zero is positive.

Nothing positive about it. It can only be considered part of the positive integers by definition. It is not a positive anything.

It devolved from there, so let's start over: Do you know that 0 is an integer?

You have a habit of saying things that are not true.

The funniest is the negative integers END at zero.

Even though zero is not a negative integer.

Go and learn the meaning of the word axiomatic.

Axioms are not proven; they are the basis on which other truths can be proven.

You're spewing nonsense you cannot back up.

You can in no way prove zero is an integer.

You don't appear to disagree with me. What exactly are you arguing with?

 

[beero1000]

beero1000 already did this once. Here it is again:

You're so infatuatiated with me you don't seem to be able to read a word I write.

I said zero is not part of the POSITIVE integers. It is not a part of that set. It is not the first or last positive integer.

Let's go on a journey to the recent past and see what happened, shall we?

You posted this:

Anything can be done by definition. You can say 0 is an integer by definition even though zero has features unlike any other integer.

I thought to myself "That's weird, it kinda sounds like you don't think 0 should be an integer". It might have just been weird phrasing though, so fine, let it slide. But then you posted this:

The integers are two separate series. The positive integers and the negatives. In both series the lowest integer is the START of the series. You cannot have a series without a start to it.

You missed zero in the integers again! So I responded:

This is your second post here where you imply that you don't think 0 is an integer. Weird.

Your responses were these strawmen implying that I said 0 is a positive integer.

Not strange at all to those capable of thinking.

Zero is not positive.

How one could think it is a positive integer is the strange thing.

I know. Very troubling when people are so lost they think zero is positive.

Nothing positive about it. It can only be considered part of the positive integers by definition. It is not a positive anything.

It devolved from there, so let's start over: Do you know that 0 is an integer?

I'd like to update my post with this comment:

Nobody will find a claim from me that zero is not an integer, though I have my doubts.

I was just suspicious before, but now I really think that untermensche doesn't know that 0 is an integer.

The roll continues.

 

[Kharakov]

Has nothing to do with the conversation. So what?

You seem to be ignoring the END part.

I could also say the series of negative integers never ends.

It does. The conventional direction is from - infinity to an end at 0 (or -1).

No infinite series has an end. That is the fucking definition.

All have a beginning.

Repeating false nonsense about infinite sequences isn't going to change that time itself doesn't have a beginning, nor the fact that infinite sequences can have ends and beginnings.

Take a section of the boundary of the Mandelbrot set, from 0 to pi on the complex plane. It is infinite in length, yet starts at 0 (.3,0) and ends at pi (-2,0). It has a specific end and beginning, yet it is infinite in length, which means there are infinite finite length sections of it.

That's just one of an infinite amount of infinite sequences that have beginnings and ends.

A series can only begin from a defined element. And it can only progress to another defined element.

A sequence of numbers can be specified to not have a beginning. Ask beero- he actually is a professional mathematician, and quite willing to impart knowledge on even the most reluctant of idiots.

 

[bilby]

You seem to be ignoring the END part.

I could also say the series of negative integers never ends.

It does. The conventional direction is from - infinity to an end at 0 (or -1).

No infinite series has an end. That is the fucking definition.

All have a beginning.

Repeating false nonsense about infinite sequences isn't going to change that time itself doesn't have a beginning, nor the fact that infinite sequences can have ends and beginnings.

Take a section of the boundary of the Mandelbrot set, from 0 to pi on the complex plane. It is infinite in length, yet starts at 0 (.3,0) and ends at pi (-2,0). It has a specific end and beginning, yet it is infinite in length, which means there are infinite finite length sections of it.

That's just one of an infinite amount of infinite sequences that have beginnings and ends.

A series can only begin from a defined element. And it can only progress to another defined element.

A sequence of numbers can be specified to not have a beginning. Ask beero- he actually is a professional mathematician, and quite willing to impart knowledge on even the most reluctant of idiots.

"You can take a horse to water, but a pencil must be lead." - Stan Laurel.

 

[ryan]

... -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 ...

Where is the beginning here?

This has only been gone over about 50 times.

The series of the positive integers begins at 1.

The series of the negative integers begins at -1. It is the same exact series as the positive integers with a negative sign in front of the number.

But none of this has any application here.

None of this applies to time or infinite time.

You said that if something has an order, then it has a beginning. But there is no beginning to the integers.

 

[untermensche]

This has only been gone over about 50 times.

The series of the positive integers begins at 1.

The series of the negative integers begins at -1. It is the same exact series as the positive integers with a negative sign in front of the number.

But none of this has any application here.

None of this applies to time or infinite time.

You said that if something has an order, then it has a beginning. But there is no beginning to the integers.

Of course there is.

The positive integers begin a 1 and the negative integers begin at -1.

ALL infinite series have a beginning.

NONE have an end.

 

[untermensche]

No infinite series has an end. That is the fucking definition.

All have a beginning.

Repeating false nonsense about infinite sequences isn't going to change that time itself doesn't have a beginning, nor the fact that infinite sequences can have ends and beginnings.

That is true.

That is why I only speak the truth.

Infinite time in the past is impossible. Infinite time never passes. It cannot be in the past.

It is not possible to crawl out of a hole infinitely deep.

ALL infinite series have a beginning. You can't show me one without one.

NO infinite series has an end. You can't show me one with one.

If you show me an infinite series you think ends that only means you don't know the difference between it's beginning and end.

 

[untermensche]

beero1000 already did this once. Here it is again:

You can't explain one thing on your own can you?

I am still waiting for the time I claimed zero was not an integer.

 

[Speakpigeon]

I'm no specialist but I believe some mathematicians insist on a so-called 'constructionist' approach to theoretical concepts. So I can sympathise with the idea that if we had to construct the set of Integers prior to being allowed to say anything about it then any talk of a set without a beginning might be ruled out. It seems true that no one human could construct the whole set of Integers starting from infinity, or starting from a non-existent beginning. You'd have to start from some arbitrary integer accepted as such, say -45010045648, or -1 if you prefer, and then count backward to get more negative integers.

Well, does it work though?

The idea would be that you could count backward up to any integer you'd be interested in. But, is that even possible? I'd say that however fast you would count those integers, there will always be at least one integer beyond your current capabilities, and obviously all integers beyond this one too. So, in practical terms, all human beings will ever be able to achieve would be a finite subset of the infinite set of Integers. Which means mathematics would have to be amputated from an entire section of studies. So, a constructionist approach will fail to produce the set of Integers as we envision it, with no good reason. So, why should we indulge the constructionist view (if that's what it is) at all?

And if we disregard the constructionist view, then there's no cogent reason to insist on saying that the beginning of the set of the negative Integer is -1 for example.

We could change our custom in this respect but I guess we follow the pattern that a finite set will be said to begin at its smallest integer and end at its biggest one, not some arbitrary value like -1. And so, naturally, an infinite past is said to have no beginning and to have an end, namely the present time. Big deal.

Some people here apparently think that's no way to talk. But that's definitely the way most people, in fact probably nearly everybody who has a modicum of expertise, will talk. And no good reason for changing our custom in this respect since no cogent argument for doing so has been presented, in all of 122 pages!
EB

I would call that  finitism rather than  constructivism. Constructivists generally accept infinite sets - what they don't like is claiming existence of an object without a way to construct it (even theoretically, or in a limit). I'm not a constructivist, but I think Phil is, so maybe he can chime in here.

Right, thanks.
I looked up your finitism link on Wiki and this quote concerning Hilbert may shed some welcome light on this obscure debate:

Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving consistency of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert's views are also associated with formalist philosophy of mathematics. Hilbert's goal of proving the consistency of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems. However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means.

It's interesting in my view because it says Hilbert couldn't in the end show his idea to be true, even though most people, including me, would probably agree it must be true in some way.

So, it highlights the, in my view, epistemological divide between concepts of the finite and the infinite. This is essential in my view. In particular, I think it probably underpins, and therefore explains to a large extent, UM's stance. It makes UM more human than his sounding off suggests.

We can even question on this basis the acceptance of the concept of infinite sets by Constructivists as being not constructivist at all, since they couldn't show how it is constructed. Those Constructivists are relapsed Realists.

Still, I think Hilbert got his distinction right between "concrete object" and "ideal object" even though we have to live without a formal proof that one can be deduced from the other.
EB

 

[Speakpigeon]

I mostly insist on a constructivist approach to mathematics, and the stuff I gave a few pages ago is constructive. In general, you are right that you start at some particular integer, and you produce the rest from some generative operations.

And no one can show how to generate any infinite set...

As beero1000 says, constructivism is distinct from finitism, though, and constructive logic generally assumes that the domain of discourse is infinite. At the same time, all objects that constructivists exhibit and talk about concretely are effectively finite.

So, Constructivists are closet-Realists about infinity. They want their constructivist cake and eat it to get the extase of infinity.

Finitists are stricter, and it is always impossible possible to interpret finitist discourse as being about a world in which there is only a finite amount of data available.

It is truly fascinating that the mathematical ship of fools should split on the reef of our performing abilities. I can do this but I can't do that. I can imagine infinity but I can't construct it.

The fundamental nature of this problem may also explain why this thread is going on and on and on, even though UM is only repeating himself ad nauseam.

I'm trying to bring closure here, and relief.
EB