The meaning of infinity

[steve_bank]

Apparently an ancient question. Theren is mathematical rigorous definitions of infinity as in calcukus and yjrtr philisophical speculations.

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


The infinity symbol
Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]....

Real analysis
In real analysis, the symbol {\displaystyle \infty } \infty , called "infinity", is used to denote an unbounded limit.[21] {\displaystyle x\rightarrow \infty } x\rightarrow \infty means that x grows without bound, and {\displaystyle x\to -\infty } x\to -\infty means the value of x is decreasing without bound.

http://mathworld.wolfram.com/Infinity.html

Infinity, most often denoted as infty, is an unbounded quantity that is greater than every real number. The symbol infty had been used as an alternative to M (1000) in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.

 

[steve_bank]

The IEE standard on digital floating point computaion. When an attmept is made to divide by zero an excption is generd in the software package. It is called infinie result.

https://en.wikipedia.org/wiki/IEEE_7...ption_handling

The five possible exceptions are:

Invalid operation: mathematically undefined, e.g., the square root of a negative number. Returns qNaN by default.
Division by zero: an operation on finite operands gives an exact infinite result, e.g., 1/0 or log(0). Returns ±infinity by default.
Overflow: a result is too large to be represented correctly (i.e., its exponent with an unbounded exponent range would be larger than emax). Returns ±infinity by default for the round-to-nearest mode.
Underflow: a result is very small (outside the normal range) and is inexact. Returns a subnormal or zero by default.
Inexact: the exact (i.e., unrounded) result is not representable exactly. Returns the correctly rounded result by default.

 

[steve_bank]

What is the definition of many? When ancient Zog ciunted mastedons and used up his fingers and toes he might have opened his arms wide and grunted a sound meaning many, uncountable.

More precisely or rigorously, the limit as x ->0 1/x is infinity. Here infinity means an unreacable or uncountable number of iterations. An asymptote. For every x there is an x+1 that can be calculated. There is no mathematical algorith or amalog or digital circuit that can divide by exactly zero.

Or a series from high school algebra. 1, 2, 3...infinity. Three terms to identify the sequence. There is no possible end to the sequence. Other than running out of paper or memory storage.

In a simple algorithm in words how would you describe the growth of i?
i=0
start
print i
i = i + 1
goto start

 

[Phil Scott]

[...]and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion.

That's incorrect. The method of exhaustion doesn't use infinitely small quantities, and the axiom which justifies the method, attributed variously to Eudoxus and Archimedes, is an axiom that explicitly asserts that there are no such quantities. The idea behind the method of exhaustion is still basically the same method we use today for integration, which again, is based on foundations that explicitly reject infinitesimals (and are thus called "Archimedean").

 

[Speakpigeon]

Apparently an ancient question. Theren is mathematical rigorous definitions of infinity as in calcukus and yjrtr philisophical speculations.

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


The infinity symbol
Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]....

Real analysis
In real analysis, the symbol {\displaystyle \infty } \infty , called "infinity", is used to denote an unbounded limit.[21] {\displaystyle x\rightarrow \infty } x\rightarrow \infty means that x grows without bound, and {\displaystyle x\to -\infty } x\to -\infty means the value of x is decreasing without bound.

http://mathworld.wolfram.com/Infinity.html

Infinity, most often denoted as infty, is an unbounded quantity that is greater than every real number. The symbol infty had been used as an alternative to M (1000) in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.

Nice thread. I didn't know that things had "meaning". So it goes off on a rather bad footing. Still, following my good counsel's advice to try charity in how I interpret people's utterances, I'll assume you meant to talk about the meaning of "infinity", i.e. the meaning of the word "infinity", which suddenly makes sense.

My initial remark would be that we're immediately into fuzzy thinking. "In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number". So, it is a bird or not? Apparently it's not a number. So, it's something else, which gets to deserve and receive treatments that somewhat look like treatments for numbers. So maybe we can divide infinity by 2, meaning perhaps that we can divide infinity in two just like we can divide a pie in two even though a pie isn't a number. Obviously, the results are not the same. We get something broadly like 2 half-pies but presumably 2 full-blown infinities rather than something like 2 half-infinities. Just assuming here since I'm a very professional layman myself when it comes to mathematics. Maybe other people could comment on that, who knows.

Also, there's what seems to me to be a very serious fuzziness in there but I don't want to talk to myself about that in front of everybody. I do it a lot anyway but I do it when I know nobody is looking.
EB

 

[Juma]

Apparently an ancient question. Theren is mathematical rigorous definitions of infinity as in calcukus and yjrtr philisophical speculations.

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


The infinity symbol
Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]....

Real analysis
In real analysis, the symbol {\displaystyle \infty } \infty , called "infinity", is used to denote an unbounded limit.[21] {\displaystyle x\rightarrow \infty } x\rightarrow \infty means that x grows without bound, and {\displaystyle x\to -\infty } x\to -\infty means the value of x is decreasing without bound.

http://mathworld.wolfram.com/Infinity.html

Infinity, most often denoted as infty, is an unbounded quantity that is greater than every real number. The symbol infty had been used as an alternative to M (1000) in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.

Nice thread. I didn't know that things had "meaning". So it goes off on a rather bad footing. Still, following my good counsel's advice to try charity in how I interpret people's utterances, I'll assume you meant to talk about the meaning of "infinity", i.e. the meaning of the word "infinity", which suddenly makes sense.

My initial remark would be that we're immediately into fuzzy thinking. "In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number". So, it is a bird or not? Apparently it's not a number. So, it's something else, which gets to deserve and receive treatments that somewhat look like treatments for numbers. So maybe we can divide infinity by 2, meaning perhaps that we can divide infinity in two just like we can divide a pie in two even though a pie isn't a number. Obviously, the results are not the same. We get something broadly like 2 half-pies but presumably 2 full-blown infinities rather than something like 2 half-infinities. Just assuming here since I'm a very professional layman myself when it comes to mathematics. Maybe other people could comment on that, who knows.

Also, there's what seems to me to be a very serious fuzziness in there but I don't want to talk to myself about that in front of everybody. I do it a lot anyway but I do it when I know nobody is looking.
EB

If you define number as a measure of size then there are many uses for infinite numbers aka transfinite numbers.

 

[phands]

The IEE standard on digital floating point computaion. When an attmept is made to divide by zero an excption is generd in the software package. It is called infinie result.

https://en.wikipedia.org/wiki/IEEE_7...ption_handling

The five possible exceptions are:

Invalid operation: mathematically undefined, e.g., the square root of a negative number. Returns qNaN by default.
Division by zero: an operation on finite operands gives an exact infinite result, e.g., 1/0 or log(0). Returns ±infinity by default.
Overflow: a result is too large to be represented correctly (i.e., its exponent with an unbounded exponent range would be larger than emax). Returns ±infinity by default for the round-to-nearest mode.
Underflow: a result is very small (outside the normal range) and is inexact. Returns a subnormal or zero by default.
Inexact: the exact (i.e., unrounded) result is not representable exactly. Returns the correctly rounded result by default.

TBH, I don't think those apply in a mathematical discussion....they are methods of preventing problems in compute hardware - from never-ending calculations to outright crashes. For instance, an implementation of a divider is repeated subtraction until the remainder of the most recent subtraction is smaller than the divisor or some fraction thereof as needed. Subtraction is slow, and repeated subtractions slower still. Division, unlike multiplication, requires a test at the end of each step to decide what to do next, so the borrow has to be distributed to all parts of a hardware divider. In the case of a divide by zero, the result of the subtraction always leaves the dividend the same, so it can never finish. Thus, hardware looks for a zero divisor to avoid an infinite(!) loop where the calculation can never end.

Floating point calculations lead to all sorts of boundary issues, and that's why the IEE standard exists...but it all has little to do with the mathematical concept of Infinity, which is NOT a number and mostly doesn't behave as one, as I said in the other thread..

 

[phands]

In a simple algorithm in words how would you describe the growth of i?
i=0
start
print i
i = i + 1
goto start

The growth is linear, so that's how I'd describe it. Again, it's an example of an infinite (compute) loop, but it won't actually continue to infinity. It will run until the value of i exceeds the capacity of the hardware registers holding the variable is exceeded. or maxed out. That value for a 64bit register in a big, but by no means infinite Integer... 9,223,372,036,854,775,807 ( Hex 7FFF,FFFF,FFFF,FFFF), assuming we're doing straight signed binary. Modern hardware will detect that the calculation has gone past the usefulness, and halt the operation with an error. Old hardware used to just keep going and restart from zero until interrupted externally.

Interestingly, Trolls in the wondrous Terry Pratchett discworld novels were notoriously slow at mental processes at room temperatures, and only counted :- "one, two, many, lots". At least until he invented the superconducting troll in an ice-house!

 

[Speakpigeon]

Nice thread. I didn't know that things had "meaning". So it goes off on a rather bad footing. Still, following my good counsel's advice to try charity in how I interpret people's utterances, I'll assume you meant to talk about the meaning of "infinity", i.e. the meaning of the word "infinity", which suddenly makes sense.

My initial remark would be that we're immediately into fuzzy thinking. "In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number". So, it is a bird or not? Apparently it's not a number. So, it's something else, which gets to deserve and receive treatments that somewhat look like treatments for numbers. So maybe we can divide infinity by 2, meaning perhaps that we can divide infinity in two just like we can divide a pie in two even though a pie isn't a number. Obviously, the results are not the same. We get something broadly like 2 half-pies but presumably 2 full-blown infinities rather than something like 2 half-infinities. Just assuming here since I'm a very professional layman myself when it comes to mathematics. Maybe other people could comment on that, who knows.

Also, there's what seems to me to be a very serious fuzziness in there but I don't want to talk to myself about that in front of everybody. I do it a lot anyway but I do it when I know nobody is looking.
EB

If you define number as a measure of size then there are many uses for infinite numbers aka transfinite numbers.

Sorry, I don't understand how that relates to what I just said. Are there infinite numbers, really?
EB

 

[Juma]

Nice thread. I didn't know that things had "meaning". So it goes off on a rather bad footing. Still, following my good counsel's advice to try charity in how I interpret people's utterances, I'll assume you meant to talk about the meaning of "infinity", i.e. the meaning of the word "infinity", which suddenly makes sense.

My initial remark would be that we're immediately into fuzzy thinking. "In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number". So, it is a bird or not? Apparently it's not a number. So, it's something else, which gets to deserve and receive treatments that somewhat look like treatments for numbers. So maybe we can divide infinity by 2, meaning perhaps that we can divide infinity in two just like we can divide a pie in two even though a pie isn't a number. Obviously, the results are not the same. We get something broadly like 2 half-pies but presumably 2 full-blown infinities rather than something like 2 half-infinities. Just assuming here since I'm a very professional layman myself when it comes to mathematics. Maybe other people could comment on that, who knows.

Also, there's what seems to me to be a very serious fuzziness in there but I don't want to talk to myself about that in front of everybody. I do it a lot anyway but I do it when I know nobody is looking.
EB

If you define number as a measure of size then there are many uses for infinite numbers aka transfinite numbers.

Sorry, I don't understand how that relates to what I just said. Are there infinite numbers, really?
EB

Yes. Both cardinal numbers (size) and ordinal numbers (order) but necessarily subtraction or division.

Google transfinite numbers, check the wiki page.

 

[Phil Scott]

Nice thread. I didn't know that things had "meaning". So it goes off on a rather bad footing. Still, following my good counsel's advice to try charity in how I interpret people's utterances, I'll assume you meant to talk about the meaning of "infinity", i.e. the meaning of the word "infinity", which suddenly makes sense.

My initial remark would be that we're immediately into fuzzy thinking. "In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number". So, it is a bird or not? Apparently it's not a number. So, it's something else, which gets to deserve and receive treatments that somewhat look like treatments for numbers. So maybe we can divide infinity by 2, meaning perhaps that we can divide infinity in two just like we can divide a pie in two even though a pie isn't a number. Obviously, the results are not the same. We get something broadly like 2 half-pies but presumably 2 full-blown infinities rather than something like 2 half-infinities. Just assuming here since I'm a very professional layman myself when it comes to mathematics. Maybe other people could comment on that, who knows.

Also, there's what seems to me to be a very serious fuzziness in there but I don't want to talk to myself about that in front of everybody. I do it a lot anyway but I do it when I know nobody is looking.
EB

If you define number as a measure of size then there are many uses for infinite numbers aka transfinite numbers.

Sorry, I don't understand how that relates to what I just said. Are there infinite numbers, really?
EB

There are transfinite cardinals and ordinals, as Juma says, but I also gave you the example of the hyperreals here. The hyperreals have, by construction, the exact same algebra as the reals, so you can take an infinite hyperreal, divide it by 2, take its reciprocal, square it, whatever, and all the normal algebraic laws hold.

 

[steve_bank]

In none of those instancesn is theer a quantifiable infinity.

The set of reals from 1 to 3 is infinite. The set of reals from 3 to 6 is infinite.

You can say the set of reals from 1 to 6 is the combination of the two infinities, but it gets you nowhere. You can derive an algebraic calculus to manipulate infinite sets, but infinity as uncountable remains.

 

[Juma]

In none of those instancesn is theer a quantifiable infinity.

The set of reals from 1 to 3 is infinite. The set of reals from 3 to 6 is infinite.

You can say the set of reals from 1 to 6 is the combination of the two infinities, but it gets you nowhere. You can derive an algebraic calculus to manipulate infinite sets, but infinity as uncountable remains.

Who, and what, are you arguing against?

 

[Speakpigeon]

In none of those instancesn is theer a quantifiable infinity.

The set of reals from 1 to 3 is infinite. The set of reals from 3 to 6 is infinite.

You can say the set of reals from 1 to 6 is the combination of the two infinities, but it gets you nowhere. You can derive an algebraic calculus to manipulate infinite sets, but infinity as uncountable remains.

Who, and what, are you arguing against?

He's just expressing his Royal displeasure at the license mathematicians take with infinity.

It's fair to say he has a point.

Show

One point. Just one point in an infinity of them.

EB

 

[Speakpigeon]

Sorry, I don't understand how that relates to what I just said. Are there infinite numbers, really?
EB

There are transfinite cardinals and ordinals, as Juma says, but I also gave you the example of the hyperreals here. The hyperreals have, by construction, the exact same algebra as the reals, so you can take an infinite hyperreal, divide it by 2, take its reciprocal, square it, whatever, and all the normal algebraic laws hold.

Sorry, I tried to understand the point of those and I do to some extent but, by and large, it still looks like poetic license to me.

Still, I don't mind the idea. Me, I think of it more intuitively, for example: 0, 1, 2, ... , ∞₃, ∞₂, ∞₁, ∞₀.

I think you should find it works.
EB

 

[Speakpigeon]

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. (...)
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one.

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

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line?!

Any real number can be determined by a possibly infinite decimal representation?!

You can't blame me if I'm confused as to whether mathematicians think of the Real numbers as integers plus a possibly a non-zero decimal part, or if they think they're somehow all the quantities in a continuum. I would have thought you'd need to choose one or the other. And this being mathematicians, I would suggest they keep with the decimal numbers. Seems we know what we're talking about there. Less metaphysical, so to speak.

How could we even prove that things like the decimal numbers map any continuum?
EB

 

[Phil Scott]

Sorry, I tried to understand the point of those and I do to some extent but, by and large, it still looks like poetic license to me.

They're justifiably how Newton and Leibniz and everyone between them and the modern real analysts understood the continuum and calculus. They're not so easily sniffed at.

Still, I don't mind the idea. Me, I think of it more intuitively, for example: 0, 1, 2, ... , ∞₃, ∞₂, ∞₁, ∞₀.

I think you should find it works.

Why would I? That's not how you do maths.

 

[phands]

[I think of it more intuitively, for example: 0, 1, 2, ... , ∞₃, ∞₂, ∞₁, ∞₀.

I think you should find it works.
EB

Unfortunately for that position, maths is often counterintuitive and paradoxical. This is often associated with the concept of infinity, such as Zeno's paradox, but not always, such as the Monty Hall problem, which many people refuse to believe because their intuition won't gibe with the fact that the proof is correct.

Others that may give pause for thought....

The Barber's Paradox.
The Two Children problem.
The Banach-Tarski property of a sphere....still makes my brain hurt.
Godel's Incompleteness.
Russell's Antinomy - a set of sets which are not elements of themselves

Maybe there's a thread in itself here?

 

[Speakpigeon]

[I think of it more intuitively, for example: 0, 1, 2, ... , ∞₃, ∞₂, ∞₁, ∞₀.

I think you should find it works.
EB

Unfortunately for that position, maths is often counterintuitive and paradoxical. This is often associated with the concept of infinity, such as Zeno's paradox, but not always, such as the Monty Hall problem, which many people refuse to believe because their intuition won't gibe with the fact that the proof is correct.

Or something else.

Others that may give pause for thought....

The Barber's Paradox.
The Two Children problem.
The Banach-Tarski property of a sphere....still makes my brain hurt.
Godel's Incompleteness.
Russell's Antinomy - a set of sets which are not elements themselves

Maybe there's a thread in itself here?

I'm fine with so-called paradoxes, thanks. I'm more interested in what exactly concept of the Reals mathematicians have.

Like, here:

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. (...)
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one.

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

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line?!

Any real number can be determined by a possibly infinite decimal representation?!

You can't blame me if I'm confused as to whether mathematicians think of the Real numbers as integers plus a possibly a non-zero decimal part, or if they think they're somehow all the quantities in a continuum. I would have thought you'd need to choose one or the other. And this being mathematicians, I would suggest they keep with the decimal numbers. Seems we know what we're talking about there. Less metaphysical, so to speak.

How could we even prove that things like the decimal numbers map any continuum?

EB

 

[Phil Scott]

You can't blame me if I'm confused as to whether mathematicians think of the Real numbers as integers plus a possibly a non-zero decimal part, or if they think they're somehow all the quantities in a continuum. I would have thought you'd need to choose one or the other. And this being mathematicians, I would suggest they keep with the decimal numbers. Seems we know what we're talking about there. Less metaphysical, so to speak.

How could we even prove that things like the decimal numbers map any continuum?
EB

I'd respond, but I honestly think it would be a waste of time for both of us.