The Measure Problem

[taeto]

3 hours ago, DannyTR said:

Can anyone give me a counter-example of Actual Infinity from maths or nature?

Are you aware that in math you have to give definitions of the things that you want to talk about? Could *you* give me "a counter-example of Wombats from maths" without knowing what one might think of as being the meaning of "wombat" in math? Maybe if we are lucky there might actually be found a research paper which introduces the definition of  a "wombat" as some kind of  mathematical structure. Maybe the paper also presents a Theorem which states that a wombat exists if and only if no tabmow exists, where a "tabmow" is another structure with a precise mathematical definition. If *you* then were to actually produce a tabmow, then you would have "a counter-example of a wombat from math". Is that the kind of situation that you have in mind? You do not really know of any definition of a mathematical object called "Actual Infinity", am I right?

Edited September 29, 2018 by taeto

[DannyTR]

15 minutes ago, taeto said:

You do not really know of any definition of a mathematical object called "Actual Infinity", am I right?

Actual Infinity is defined here:

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

In maths actual infinity is the infinite set concept (whereas potential infinity is closest to the limit concept).

 

 

 

[taeto]

1 minute ago, DannyTR said:

Actual Infinity is defined here:

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

      No it is not. Where on that page does one encounter such a definition which enables one to talk about "Actual Infinity" in mathematical language?

[Carrock]

7 minutes ago, taeto said:

      No it is not. Where on that page does one encounter such a definition which enables one to talk about "Actual Infinity" in mathematical language?

From   https://en.wikipedia.org/wiki/Axiom_of_infinity linked from the previous reference:

Quote

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

 

Edited September 29, 2018 by Carrock

[DannyTR]

7 minutes ago, Carrock said:

From   https://en.wikipedia.org/wiki/Axiom_of_infinity 

As far as I’m concerned the axiom of infinity is provably wrong. The definition of a set in set theory is also wrong (polymorphism).

 - All sets should have a natural number as cardinality (sets after all contain complete objects)

 - but there is no natural number that is larger than all other natural numbers

 - so the infinite set does not exist, axiom of infinity is wrong 

[taeto]

8 minutes ago, Carrock said:

From   https://en.wikipedia.org/wiki/Axiom_of_infinity linked from the previous reference:

The inductive set is familiar. How do we get from there and to a definition of "Actual Infinity"?

[studiot]

7 minutes ago, DannyTR said:

As far as I’m concerned the axiom of infinity is provably wrong.

This is just nonsense.

You don't prove an axiom 'wrong'.

Either the system of deductions constructed using that axiom, along with any other predefined ones, is self consistent or it is not.

But either way it says nothing about the relationship between that axiom (or the others) and other introduced material that has not been defined.

You have introduced extraneous, undefined material.

 

You have also asked a question, apparently leading to division by zero

"How many points are there in a line segment?"

 

Let me turn this round and ask

 

How many (distinguishably different) lines may be drawn through a given point?

 

Edited September 29, 2018 by studiot

[taeto]

4 minutes ago, DannyTR said:

The definition of a set in set theory is also wrong (polymorphism).

That is pretty hilarious. I wasn't sure before that you are just trolling. Just for fun though this question: if the objects of set theory should not be called "sets", then what is your alternative suggestion?

[Strange]

11 minutes ago, DannyTR said:

but there is no natural number that is larger than all other natural numbers

That is the reason that the axiom of infinity is, uhm ... axiomatic. 

[Carrock]

Everything I was going to post has been posted while I was composing my response...

except which is 'Actual Infinity' supposed to be?

The number of integers, the number of points on a line... etc

[DannyTR]

10 minutes ago, Strange said:

That is the reason that the axiom of infinity is, uhm ... axiomatic. 

An axiom is meant to be a self evident truth.

The axiom of infinity is a self evident falsehood.

That falsehood has made its way from the abstract study of sets into the non-abstract study of the universe and IMO is causing much confusion and time wasting.

 

[taeto]

19 minutes ago, Carrock said:

Everything I was going to post has been posted while I was composing my response...

except which is 'Actual Infinity' supposed to be?

The number of integers, the number of points on a line... etc

It is Saturday evening here; everyone has a lot of time on their hands, sorry

DannyTR already argued why there cannot be any  number larger than all other numbers, which, assuming the total ordering of numbers, is the same as saying that there has to be infinitely many numbers. By inference he is not disputing that fact.

A, ehm, mathematically inclined person might speak of the "cardinality" of the set of integers, the set of points on a line. Rarely of the "number". But you hint at the possibility that the OP is thinking about this "Actual Infinity" as a number?  

 

9 minutes ago, DannyTR said:

That falsehood has made its way from the abstract study of sets into the non-abstract study of the universe and IMO is causing much confusion and time wasting.

Q.E.D. as witnessed by this thread: OP's confusion and everyone else's time wasted.

Edited September 29, 2018 by taeto

[Strange]

Obviously one can use a form of set theory  where there are no infinite sets. Presumably, that set theory has limitations that limit its use (like not being able to argue about the set of natural numbers or reals)? Hence the introduction of the axiom of infinity. 

[DannyTR]

46 minutes ago, taeto said:

mathematically inclined person might speak of the "cardinality" of the set of integers, the set of points on a line. Rarely of the "number". But you hint at the possibility that the OP is thinking about this "Actual Infinity" as a number?  

Yes you can picture Actual Infinity in your mind by imagining things as above, but that does not mean it exists mathematically or in the real world. 

Actual Infinity is a self contradictory concept: that’s why it makes our heads hurt to think of it.

And when you try to visualise Actual Infinity, it is always in the form of a potential infinity anyway (such as imagining zooming in on a line and seeing more and more points).

The cardinality of the set of natural numbers is not a number by any reasonable definition. For example what other number can you add to and not change!

A point has size 0, so there are (line length) / 0 = UNDEFINED points on a line.

 

Edited September 29, 2018 by DannyTR
Typo

[studiot]

1 hour ago, DannyTR said:

Actual Infinity is defined here:

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

In maths actual infinity is the infinite set concept (whereas potential infinity is closest to the limit concept).

I have already pointed out that this usage is outdated and non-mainstream and Wikipedia itself accepts this.

Writ large at the top of a dead end link, except for historical guff about Ancient Greeks.

Quote

Wikipedia

This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Sections consisting of quotations must be rewritten in prose style. Use footnotes for referencing. Please help improve this article if you can. (June 2010) (Learn how and when to remove this template message)

But what Wikipedia doesn't say is

When are you going to answer my questions?

Edited September 29, 2018 by studiot

[Strange]

1 minute ago, DannyTR said:

Actual Infinity is a self contradictory concept: that’s why it makes our heads hurt to think of it.

Maybe that is because the concept is so poorly defined as to be meaningless. Can you provide a meaningful definition in your own words - not copying from that Wikipedia page you keep linking to (which I’m sure you don’t actually understand).

3 minutes ago, DannyTR said:

The cardinality of the set of natural numbers is not a number by an reasonable definition.

I think everybody knows that. So how is it relevant or useful to say it?

 

[DannyTR]

1 minute ago, Strange said:

Maybe that is because the concept is so poorly defined as to be meaningless. Can you provide a meaningful definition in your own words - not copying from that Wikipedia page you keep linking to (which I’m sure you don’t actually understand).

I think everybody knows that. So how is it relevant or useful to say it?

 

I’m not sure I can define Actual Infinity directly. I can define potential infinity as the results of repeated interations of the same process. Then actual infinity is the result of carrying on these iterations for a unlimited period of time. But I think that is a circular definition: ‘unlimited’=‘infinity’.

Actual Infinity is difficult to define because it does not exist exist mathematically or in nature; just exists in our heads as a (flawed) concept.

Something that we cannot cleanly define and is not a rational concept, does not have a place in scientific theories IMO. 

[Ghideon]

2 minutes ago, DannyTR said:

Actual Infinity is difficult to define because it does not exist exist mathematically or in nature; just exists in our heads as a (flawed) concept.

Something that we cannot cleanly define and is not a rational concept, does not have a place in scientific theories IMO. 

I'm no math expert but above seems like a sound statement about Actual Infinity. Maybe that is why Actual Infinity is somewhat outdated and replaced by newer definitions?

[Strange]

13 minutes ago, DannyTR said:

I’m not sure I can define Actual Infinity directly.

So you have an entire thread about your quasi-religious beliefs about it but you are unable to say precisely what it is. 

I think this tells us all we need to know. 

15 minutes ago, DannyTR said:

Actual Infinity is difficult to define because it does not exist exist mathematically or in nature; just exists in our heads as a (flawed) concept.

So you are arguing for the non existence of a meaningless concept. 

15 minutes ago, DannyTR said:

Something that we cannot cleanly define and is not a rational concept, does not have a place in scientific theories IMO. 

Then why do you keep banging on about it. Note that is only you. No one else ever brings it up. 

So you are arguing against an undefined straw man. Why?

 

[DannyTR]

5 minutes ago, Ghideon said:

I'm no math expert but above seems like a sound statement about Actual Infinity. Maybe that is why Actual Infinity is somewhat outdated and replaced by newer definitions?

Well it’s old but not really outdated. Aristotle came up with potential/actual infinity 2000 years ago and no one has come up with a better explanation of infinity since then.

Set theory merely defines Actual Infinity to exist in the form of a complete infinite set, by way of axiom and without proving anything.

Calculus uses the limit concept; approaching but not reaching Actual Infinity.

So maths sort of ducks the issue of defining Actual Infinity. Not surprising really; it’s a mad concept, hard to define.

 

[Carrock]

1 hour ago, taeto said:

A, ehm, mathematically inclined person might speak of the "cardinality" of the set of integers, the set of points on a line. Rarely of the "number". But you hint at the possibility that the OP is thinking about this "Actual Infinity" as a number?

I was. Or wondering if he considered the cardinality of aleph-1 a bit nearer "Actual Infinity" than aleph-null and so on. Talking about cardinality seems a bit abstract for this thread.

 

23 minutes ago, DannyTR said:

I’m not sure I can define Actual Infinity directly. I can define potential infinity as the results of repeated interations of the same process.Then actual infinity is the result of carrying on these iterations for a unlimited period of time.

Do you believe that time will 'stop' at a specific moment in the future then? If it doesn't then the universe is becoming "Actually Infinitely" old. Which is less than infinitely old...

 

Slow posting again.....

Edited September 29, 2018 by Carrock
Grammar - change of tense

[DannyTR]

1 minute ago, Carrock said:

I was. Or wondering if he considered the cardinality of aleph-1 a bit nearer "Actual Infinity" than aleph-null and so on. Talking about cardinality seems a bit abstract for this thread.

Do you believe that time will 'stop' at a specific moment in the future then? If it doesn't then the universe will eventually become "Actually Infinitely" old. Which is less than infinitely old...

Actual Infinity does not behave like a quantity or even an object. What other thing can you add to without changing?

I’m Eternalist and a Finitist so I have to believe time has a end.

[studiot]

14 minutes ago, DannyTR said:

I’m Eternalist and a Finitist so I have to believe time has a end.

Since you won't answer questions and bend over backwards to contradict yourself I think this has run its weary course.

 

reported.

Edited September 29, 2018 by studiot

[Strange]

16 minutes ago, DannyTR said:

no one has come up with a better explanation of infinity since then.

I think you will find you are wrong there. It has been put on a firm mathematical basis. Your faith just won’t let you understand that. 

13 minutes ago, DannyTR said:

I’m Eternalist and a Finitist so I have to believe time has a end.

How can an “eternalist” think time has an end? That sounds like a contradiction. I think you need a new name for your religion. 

[taeto]

38 minutes ago, DannyTR said:

Actual Infinity is difficult to define because it does not exist exist mathematically or in nature; just exists in our heads as a (flawed) concept.

Then why do you want to discuss about it all the time? Noone else has an idea what it is about, and neither do you.

Do you happen to know about the theory of "True Arithmetic"? It has only natural numbers 1,2,3... in it, and all kinds of true theorems on addition and multiplication of those numbers. It is also known as the standard model of the natural numbers. There are no infinite objects in this theory. It is a part of mathematics, but would you approve of it nonetheless?