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

[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

 

[Phil Scott]

Well, this is embarrassing for one of us.

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.

The definition I gave in code has the integers defined in three distinct parts: there is 0; there are the positive integers starting from 1 and going upwards; and finally, there are the negative integers starting from -1 and going downwards.

 

[untermensche]

Post whatever irrelevancies you want.

They are just to hide the fact you are incapable of thought. Incapable of forming a logical argument.

Nothing can move FROM infinity.

 

[beero1000]

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.

 

[Phil Scott]

Post whatever irrelevancies you want.

They are just to hide the fact you are incapable of thought. Incapable of forming a logical argument.

Nothing can move FROM infinity.

I'm actually trying to defend your position, here. That you knee-jerk attack everyone on this thread should be evidence that this thread is a colossal waste of time.

 

[untermensche]

The definition I gave in code has the integers defined in three distinct parts: there is 0; there are the positive integers starting from 1 and going upwards; and finally, there are the negative integers starting from -1 and going downwards.

Your definition of the integers was noted.

Is zero positive?

If not it is not a positive integer or part of the series of positive integers.

It is part of neither the series of positive or negative integers.

- - - Updated - - -

Post whatever irrelevancies you want.

They are just to hide the fact you are incapable of thought. Incapable of forming a logical argument.

Nothing can move FROM infinity.

I'm actually trying to defend your position, here. That you knee-jerk attack everyone on this thread should be evidence that this thread is a colossal waste of time.

No. I appreciate your effort. That comment was not directed at you. People seem to be posting fast and furious.

 

[Phil Scott]

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 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.

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

 

[untermensche]

All of this is off point and a tangent.

The only salient point to note is that no series can begin at infinity and progress from there. Infinity is not a place or an amount or a number. It is an imaginary concept. Nothing can progress from it.

You cannot move from the undefined to the defined. You can only move from something defined to some other thing defined.

 

[Kharakov]

The set of integers does not begin or end, and really has little to do with a timeline, other than being an easy to understand (for educated non-morons) example of something that does not have a start or an end.

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.

Unless you have a series without a start. Then you just say something like "...., -1000, -999,-998...", arbitrarily picking a location in the series to indicate the property of the series.

Where you pick a location to demonstrate properties of the series doesn't really matter. ...5,6,7... is the same as ....-7, -6, -5....


Of course, if you have a well defined origin (you're measuring from a zero point that you define, like defining AD/BC), then ...5,6,7... = 12 + [... -7, -6, -5...].

 

[Kharakov]

The only salient point to note is that no series can begin at infinity and progress from there. Infinity is not a place or an amount or a number. It is an imaginary concept. Nothing can progress from it.

You are right. Something cannot begin at -infinity (or infinity), because they are not set points. A beginning is a specific point in time. Something without a beginning (something infinite, like existence), doesn't have a beginning. <--what is it that he did in the last statement, since he did not include all the assumed premises that everyone seems to know?

 

[untermensche]

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.

Unless you have a series without a start. Then you just say something like "...., -1000, -999,-998...", arbitrarily picking a location in the series to indicate the property of the series.

That is a series with a start. You are starting the series at -1000.

Nothing can start at "...".

That is not a thing anything can start at.

 

[George S]

"Beginning"s and "end"s are all a bit informal though, and I prefer to actually go through the details and let formalisation speak for itself. And when you get serious about that, and build the actual models in a formal calculus, you end up with the sort of thing I describe above (which can be entered with little modification in a theorem prover for constructive type theory).

Note that in the code I provide, I do define the standard ordering of the integers, and have it that -2 < -1. I don't anywhere talk about beginnings or ends, though. I instead mention the idea of a base of induction/recursion, which in the case of models of integers, will be with 0, 1 and -1.

As for untermensche's confusion, that's some off the board stuff which I'm not interested in trying to sort out. I love the fact that you folk are still trying though.

I wasn't trying to formalize it - though you could very easily define the first element and last element of a toset (and people do, even though they're usually called the minimum and maximum). The constructive models don't affect that, which is why I was saying that it's important to keep in mind the difference between 'are ordered this way' and 'are enumerated this way'. I think the distinction is much more evident in the rationals - if you wanted to enumerate them formally you certainly wouldn't do it in their standard order.

The set of integers does not begin or end, and really has little to do with a timeline, other than being an easy to understand (for educated non-morons) example of something that does not have a start or an end.

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.

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

Technically 0 is an integer. It is not, however, a natural number.

 

[Phil Scott]

All of this is off point and a tangent.

The only salient point to note is that no series can begin at infinity and progress from there. Infinity is not a place or an amount or a number. It is an imaginary concept. Nothing can progress from it.

You cannot move from the undefined to the defined. You can only move from something defined to some other thing defined.

The exact logical moves one can make are a point of controversy in the philosophy of mathematics, especially when the infinite rears its head. Logicians, mathematicians and philosophers of mathematics have all attempted to clearly specify what the valid logical moves are, and some of the restricted sets of rules form what is called "constructivism" and "finitism". Those restricted rule sets, as I suggest, might well be sympathetic to your ideas, whilst clarifying them into unambiguous formalism.

 

[Kharakov]

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.

Unless you have a series without a start. Then you just say something like "...., -1000, -999,-998...", arbitrarily picking a location in the series to indicate the property of the series.

Where you pick a location to demonstrate properties of the series doesn't really matter. ...5,6,7... is the same as ....-7, -6, -5....

Of course, if you have a well defined origin (you're measuring from a zero point that you define, like defining AD/BC on the western calendar), then ...5,6,7... = 12 + [... -7, -6, -5...].

Nothing can start at "...".

Exactly. It's not a starting point, it implies that the series doesn't have a start. You're getting it!

 

[Phil Scott]

Technically 0 is an integer. It is not, however, a natural number.

Opinion is divided, with logicians and computability theorists tending to include "0" as a natural number, and number theorists tending to exclude it. I find it amusing that, when I did my maths degree, we got mathematical logic and number theory presented in the same course. In the mathematical logic stream, we were told that 0 is in N. In the number theory stream, we were told that 0 is not in N.

 

[beero1000]

Technically 0 is an integer. It is not, however, a natural number.

Opinion is divided, with logicians and computability theorists tending to include "0" as a natural number, and number theorists tending to exclude it. I find it amusing that, when I did my maths degree, we got mathematical logic and number theory presented in the same course. In the mathematical logic stream, we were told that 0 is in N. In the number theory stream, we were told that 0 is not in N.

For good reason!

Of course, 0 being an integer is not really a point of contention, no matter how much untermensche protests.

 

[untermensche]

For good reason!

Of course, 0 being an integer is not really a point of contention, no matter how much untermensche protests.

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.

 

[untermensche]

Exactly. It's not a starting point, it implies that the series doesn't have a start. You're getting it!

The series you presented most definitely had a beginning.

It began at -1000 and grew larger from there.

There is no such thing as a series with no beginning.

 

[George S]

Opinion is divided, with logicians and computability theorists tending to include "0" as a natural number, and number theorists tending to exclude it. I find it amusing that, when I did my maths degree, we got mathematical logic and number theory presented in the same course. In the mathematical logic stream, we were told that 0 is in N. In the number theory stream, we were told that 0 is not in N.

For good reason!

Of course, 0 being an integer is not really a point of contention, no matter how much untermensche protests.

Sure.

When I got my math degree (1968) some professor made a point that the non-negative integers include 0 but the negative integers do not. I don't remember the exact context, though. If zero is not an integer then Z = (...,-3,-2,-1,1,2,3,...) which seems wrong, somehow.

And, of course, I, myself, tried to tell our untermensche that he is correct. Counting beginning at the "..." doesn't work.

The claim that time may not only be infinite in the future time direction but in the past time direction is a case of logically possible.

And, yes, untermensche, it is easy to wonder whence this infinite timeline. You want it to begin to exist. To have a starting point, and since it doesn't then it cannot exist. This rests on the assumption that all that exists begins to exist at some point in time.

Another logical possibility is that reality has a beginning but no end. It expands until it is so dilute that it is indistinguishable from nothing at all. But here and there there are unbound quarks millions of light years from its nearest neighbor quark.

 

[ryan]

What if time is a dimension? If it is, try to think of time geometrically. What if time is infinitely dense and "inflated" perpendicularly (like perpendicularly to the direction that the consciousness traverses it or just perpendicular to itself) due to quantum fluctuations, then in this case infinite time could have occurred instantaneously. As far as I know, there nothing in cosmology that would say that this couldn't happen.

Time is that which allows events to happen.

So infinite time implies infinite events.

And infinite events could not have already "occurred". By definition.

Think of a point that gets stretched from a single dimension to 2 dimensions. If it's continuous, any infinitesimal points that suddenly appear in this new 2d string will never move past any measurable distance. There will be infinite time (or space whatever you want to label it ) behind them and infinite time in front of them (unless they are in front in the "pilot position") even though the stretch begins 2d instantaneously.