We see sand on the beach and say they are countless or infinite number of sand particles and then take a handful of sand which too have a countless number. It creates a paradox.

A set A is called finite (with n members) if there is a 1-1 correspondence between the standard collection { integers x with 0 <= x < n } and A.
The simplest definition of infinity (for sets) is: "not finite". This leaves room for denial (there is nothing that is infinite) or a philosophical attitude of "potential infinity" (the act of forming the set of all integers cannot be completed, but one can collect any desired finite part).
However, usual axiom systems of set theory like ZFC or NBG impose the existence of an infinite set, and thereby the existence of the set of all natural (ordinal) numbers..
Modern Mathematics does not borrow "existence" from the reality we live in. That's a fair attitude (does "two" exist?). Any dispute on whether "infinity really occurs" either links to philosophy or physics, or it is a technical mathematical discussion (find a reasonable and workable axiom system that does not involve infinity).

Q1. A set without bound. Mind, there are different classes of infinity.

Q2. A beach, or a handful of sand, has a finite number of grains on it. There are more grains on the beach then in the handful. There is no paradox in my opinion.

It is an abstract concept of "what has no end", ie it is difficult if not impossible at all to figure it out with our finite oriented brains.
Notice that this applies to both ends: what is infinity, what is infinitesimal, what is continuum...
Actually quantum mechanics to some extent in a very particular view solve one problem, since all can be counted in integer form.

Aug 10, 2013

Manish Kumar · National Institute of Technology, Hamirpur

@ Vitaly Voloshin: I know it is wrong to say infinite numbers. Infinity is only a concept. But how does mathematics define it?
I would like to discuss one story .... The great Indian mathematician Dr. S. Ramanujan once asked asked his class teacher while he was in 6th standard that if 6 fruits are distributed among zero persons then what it would be to each??? The answer is infinity from the math but is it possible?? Can we think of it??

@Manish..... stars you see in the night sky are infinite in number but you can count them, but the black dense sky in the background is also infinite but are not countable. To a layman, we can approach infinity as something which are in one way countable and in the other way not countable. But both are not finite. To a Mathematician, though, this metaphor is very well known and he can even compare these two infinities, uncountable infinite is bigger than countable infinite [ and believe me, as a student when I read this comparison of infinites, I was just awe-struck.].

What I think nothing is infinite. Everything is countable if there is much time and resources to do it. We call something infinite if we are unable to count it.

At first approximation, one may view "infinity" as a bit of a shorthand for describing a cardinality that is at least as large as anything finite. But even that sounds less clear on second thought, because you would then need to define what "finite" is and what "at least as large" means. I would say that to describe infinity or finiteness without going in circles, one needs the notions of a set, subset, and a 1-1 correspondence (bijection).

So then, a set is infinite if it is in a 1-1 correspondence with at least one of its nonempty proper subsets (i.e. a subset that is not the empty set nor the whole set under consideration). And then one can say that a set is finite if it is not infinite (as defined immediately above).

So finiteness is actually harder to define than infinity, believe it or not. (Don't believe it? Use _only_ the definitions in the previous paragraph (and not your intuition) to prove that if M and N are finite, then M+N is also finite. That's after you have defined the addition using only the definition above.) For example, finite numbers are usually thought of as counting (or somehow measuring) the number (or quantity) of something. But take, for example, the number 10^(10^10). It is probably greater than the number of elementary particles in our universe and the life of the universe in the smallest measurable time units (if not, just consider 10^(10^(10^10)), that should suffice). So that number does not count or otherwise measure anything in existence at any time at all, it is a pure abstraction, and yet it is easily written and classified as finite.

As for the "grains of sand on the beach", etc., this has more to do with linguistics than math. "Countless grains of sand" is just a hyperbole, we don't really mean that they can never counted as part of any controlled scientific experiment.

This just goes to show that integers as represented by, say, notches on wood are different objects than integers in exponential notation (written as decimal, binary, etc. numbers).

Usually the term "infinity" is used in two sense in mathematics . First in the sense of very large ( i.e. as x approaches to zero, 1/x approaches to infinity) and secondly in the sense of "not defined or Undefined" (i.e. 1/0, tan $\frac{\pi}{2}$ etc. )

Set theory actually deals with infinite numbers. There are even two different kinds of infinite numbers: ordinal and cardinal numbers, or short ordinals and cardinals. The ordinals are defined using order.
The first ordinals are the natural numbers. Then there is the first ordinal that is larger than all the natural numbers. This one has a successor, and that successor has a successor and so on. So in some precise sense, the ordinals allow to continue counting as long as you wish.
Every ordinal number is constructed as a set, namely the set of its predecessors.
So 0 is just the empty set, as 0 has no predecessors (among the natural numbers)
1 is the set with the single element 0. 2 has the elements 0 and 1 and so on.
The first infinite ordinal is just the set of all natural numbers.

The second type of infinite numbers, the cardinals, are technically defined as special ordinals. The cardinals measure sizes of sets. The smallest infinite size of a set is the size of the set of natural numbers. However, for every set there is a larger set.
Therefore for every cardinal there is a larger cardinal.

The connection between cardinals and ordinals is the following: It can be shown that
for every set there is an ordinal of the same size. Now we can assign to every
set the smallest ordinal of the same size. These smallest ordinals are the cardinals.

Note that infinite ordinals and cardinals are abstract concepts.
Hence it is a philosophical question whether or not they exist.

A set A is called finite (with n members) if there is a 1-1 correspondence between the standard collection { integers x with 0 <= x < n } and A.
The simplest definition of infinity (for sets) is: "not finite". This leaves room for denial (there is nothing that is infinite) or a philosophical attitude of "potential infinity" (the act of forming the set of all integers cannot be completed, but one can collect any desired finite part).
However, usual axiom systems of set theory like ZFC or NBG impose the existence of an infinite set, and thereby the existence of the set of all natural (ordinal) numbers..
Modern Mathematics does not borrow "existence" from the reality we live in. That's a fair attitude (does "two" exist?). Any dispute on whether "infinity really occurs" either links to philosophy or physics, or it is a technical mathematical discussion (find a reasonable and workable axiom system that does not involve infinity).

Q1. A set without bound. Mind, there are different classes of infinity.

Q2. A beach, or a handful of sand, has a finite number of grains on it. There are more grains on the beach then in the handful. There is no paradox in my opinion.

I like Dedekind's definition: A set X is infinite if there exists a bijection (one-to-one mapping) between X and some proper subset of X.
Of course, one can give an equivalent definition: A set X is infinite if there exists an injection from the integers into X. I like Dedekind's better because he doesn't appeal to any particular infinite set in defining "infinite set."

One way to think of infinity is in terms of an algorithm which can produce sets of output of limitless size. For example, beginning with 0, 1 and the operation of addition, one can produce the sets {0,1}, {0,1,2}, {0,1,2,3}, ...and there is clearly no limit to the size of the sets in this sequence.

According to Encyclopedia Britannica: “Infinity is the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1657. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical.”
• In mathematics: ‘Minus infinity’ (–∞) is less than any real number, and ‘plus infinity’ (+∞) is greater than any real number". But, a finite number like 1/3 written as a decimal number 0.3333333... has infinite 3s after decimal zero (0).
• In Physics: According to Albert Einstein, a uniform space can be curved like a sphere and comprise a universe that is finite in volume. But, the density of the mass at the center of a simple black hole, with finite mass, is infinitely large.
• In Metaphysics: Infinity is relates to eternity.

Just a side remark , to take a break from the heavy-going thoughts - in Sanskrit, the usual word for infinity is 'ananta' .
The interesting thing is that ananta is actually 2 words - "an-" , a prefix meaning 'absence of ' and directly cognate with both our English prefix 'un-' and our word 'no', and 'anta', meaning a boundary or an end and directly akin to the English word 'end' : ananta means, both etymologically and literally, 'un-end' or 'no-end' .
(Delve a bit in the etymology of modern English, and it soons appears that its closeness with ancient Sanskrit is stunning)

Infinity (late 14c.) – derived from Old French 'infinité' ”infinity; large number or quantity” (13c.), which derived from Latin 'infinitatem' (nominative 'infinitas') ”boundlessness, endlessness”. Latin 'infinitas' itself is calqued (word-for-word) from the Greek word 'apeiria' "infinity", from 'apeiros' (άπειρος = a+peiros) "endless."

Nebi, in Latin in-finity is no-'finis', with finis originally meaning a boundary, a marker in the ground, a wooden post set in the ground - from an Indo-European word (proto-Sanskrit) which also gave the word 'dike' in English: for dwellers of low-lying lands near the sea , such as the early West Germanic tribes, the dike is the boundary.

Sep 2, 2013

Manish Kumar · National Institute of Technology, Hamirpur

What if I distribute 'zero' of something to 'n' number of people??? Math says it is 'infinite of that thing given to each' but is it really possible??

Chris, according to “Webster’s Encyclopedic Unabridged Dictionary of the English Language”: infinity (1350-1400) < ME (Middle English) infinite < L (Latin) infinitas, equivalent to in- IN- (akin to AN-, A-, UN-) + fini(s) boundary (also: end, utmost limit, highest point) + -tas –TY.

Nebi,
Latin comes from the common Indo European source of most European languages. I studied Sanskrit, and find it often extremely enlightening to find the common Ur-word source to modern European words . Latin or Greek or Germanic did not spontaneously arise ex nihilo but came from a much older ultimate source, common to most modern separate European families of languages, which is missed if you go much further back in time than 1 or 2 thousand years ago.
Examples ?
Candidate is ultimately the same word as hunger. Why ? The answer is fascinating.....
or : Hundred is 2 words, a first element stemming from the same Ur-source that gave the word cent, and the second element red from the selfsame ultimate source that yielded ratio: hundred is 'cent-ratio', i.e. the number one hundred.
Thousand, the 'big hundred' - the first element means big, as in thumb (the big finger), or as in tumor, or tumescence in other families of languages than Anglo-Saxon..
Weihen, to hallow or consecrate in German, is ultimately the same word as 'victim': Why?

Going back to only Germanic or only Latin or only Greek does not enlightenin the least, provides no clues to why and how the tongue developed, and explains nothing. To understand whence the words came and why, you have to go back to proto-Sanskrit and even sometimes to the Merritt's Ruhlen Nostratic. To do that is endlessly enlightening.

Modern European etymologies bear the marks of ancient wars, catastrophes, climate, geography, superstitions, religions, hunts, and so on: languages do not lie but constitute a fascinating repository of mankind's history. Again, explaining an etymology by Old English or Latin does not explain anything, and if you stop there a whole body of knowledge is not even scratched at.

Chris,
I agree with you that Sanskrit word 'ananta' < ‘an+anta’ means ‘no-end’, but the English word ‘infinity’ (1350-1400) certainly doesn’t come directly from Sanskrit ‘ananta’.

Nebi,
Maybe you should read posts more carefully???

Infinite, as I indicated , comes from Latin in- (cognate with un- and an-) and then a word (finis) originally meaning a wooden post stuck in the ground and marking a boundary, cognate with the Germanic word 'dike' meaning a fortified edge of water.

Infinity is always tricky, e.g. the infinite aleph numbers are all infinite but have been called 'transfinite' rather than infinite because there is a clearly demonstrable hierarchy of sizes of infinity within them (with any aleph(N+1) being infinitely larger than aleph(N).

(Their separate infinite sizes are by the way called 'cardinality' rather than size, and 'stronger' rather than larger', because we're uneasy about speaking of the different sizes of infinity, but it's just semantics that does not change anything to the underlying reality.)

It's interesting that errors can often be routinely made when dealing with infinities. For instance, the famous 'Hilbert Hotel' scenario is often used in misleading ways: if you are going to use the usual scenario whereby a first arriving guest moves into the first (occupied) room, who moves on to the second one and so on ad infinitum, the scenario actually does not work the way it is intended to, because it takes an infinite time for the guests to occupy the hotel under that scenario: the scenario has only transformed one infinity (to wit, a second infinity of number of guests) by another infinity which pops up elsewhere ( i.e., infinite time)

Interestingly, it seems relatively easy to find an infinity even larger (stronger) than aleph(∞). Infinity is always open-ended, uncircumscribable.

## Popular Answers

Marcel Van de Vel· VU University AmsterdamJames Garry· Red Core Consulting ltd.Q1. A set without bound. Mind, there are different classes of infinity.

Q2. A beach, or a handful of sand, has a finite number of grains on it. There are more grains on the beach then in the handful. There is no paradox in my opinion.

## All Answers (31)

Alok Goel· ETH ZurichRaffaele Ragone· Second University of NaplesSamuel Arba Mosquera· SCHWIND eye-tech-solutions GmbH & Co. KGNotice that this applies to both ends: what is infinity, what is infinitesimal, what is continuum...

Actually quantum mechanics to some extent in a very particular view solve one problem, since all can be counted in integer form.

Manish Kumar· National Institute of Technology, HamirpurI would like to discuss one story .... The great Indian mathematician Dr. S. Ramanujan once asked asked his class teacher while he was in 6th standard that if 6 fruits are distributed among zero persons then what it would be to each??? The answer is infinity from the math but is it possible?? Can we think of it??

Arghya Bandyopadhyay· Khalisani CollegeAbdullah Waseem· Bilkent UniversityAlexander Burstein· Howard UniversitySo then, a set is infinite if it is in a 1-1 correspondence with at least one of its nonempty proper subsets (i.e. a subset that is not the empty set nor the whole set under consideration). And then one can say that a set is finite if it is not infinite (as defined immediately above).

So finiteness is actually harder to define than infinity, believe it or not. (Don't believe it? Use _only_ the definitions in the previous paragraph (and not your intuition) to prove that if M and N are finite, then M+N is also finite. That's after you have defined the addition using only the definition above.) For example, finite numbers are usually thought of as counting (or somehow measuring) the number (or quantity) of something. But take, for example, the number 10^(10^10). It is probably greater than the number of elementary particles in our universe and the life of the universe in the smallest measurable time units (if not, just consider 10^(10^(10^10)), that should suffice). So that number does not count or otherwise measure anything in existence at any time at all, it is a pure abstraction, and yet it is easily written and classified as finite.

As for the "grains of sand on the beach", etc., this has more to do with linguistics than math. "Countless grains of sand" is just a hyperbole, we don't really mean that they can never counted as part of any controlled scientific experiment.

Alexander Burstein· Howard UniversitySudhir Kumar Pradhan· Sofia University "St. Kliment Ohridski"Stefan Geschke· University of HamburgThe first ordinals are the natural numbers. Then there is the first ordinal that is larger than all the natural numbers. This one has a successor, and that successor has a successor and so on. So in some precise sense, the ordinals allow to continue counting as long as you wish.

Every ordinal number is constructed as a set, namely the set of its predecessors.

So 0 is just the empty set, as 0 has no predecessors (among the natural numbers)

1 is the set with the single element 0. 2 has the elements 0 and 1 and so on.

The first infinite ordinal is just the set of all natural numbers.

The second type of infinite numbers, the cardinals, are technically defined as special ordinals. The cardinals measure sizes of sets. The smallest infinite size of a set is the size of the set of natural numbers. However, for every set there is a larger set.

Therefore for every cardinal there is a larger cardinal.

The connection between cardinals and ordinals is the following: It can be shown that

for every set there is an ordinal of the same size. Now we can assign to every

set the smallest ordinal of the same size. These smallest ordinals are the cardinals.

Note that infinite ordinals and cardinals are abstract concepts.

Hence it is a philosophical question whether or not they exist.

Marcel Van de Vel· VU University AmsterdamJames Garry· Red Core Consulting ltd.Q1. A set without bound. Mind, there are different classes of infinity.

Q2. A beach, or a handful of sand, has a finite number of grains on it. There are more grains on the beach then in the handful. There is no paradox in my opinion.

Kerry Soileau· Louisiana State UniversityOf course, one can give an equivalent definition: A set X is infinite if there exists an injection from the integers into X. I like Dedekind's better because he doesn't appeal to any particular infinite set in defining "infinite set."

Yogendra Singh· IPS Group of CollegesKerry Soileau· Louisiana State UniversityNebi Caka· University of Prishtina• In mathematics: ‘Minus infinity’ (–∞) is less than any real number, and ‘plus infinity’ (+∞) is greater than any real number". But, a finite number like 1/3 written as a decimal number 0.3333333... has infinite 3s after decimal zero (0).

• In Physics: According to Albert Einstein, a uniform space can be curved like a sphere and comprise a universe that is finite in volume. But, the density of the mass at the center of a simple black hole, with finite mass, is infinitely large.

• In Metaphysics: Infinity is relates to eternity.

H Chris Ransford· Karlsruhe Institute of TechnologyThe interesting thing is that ananta is actually 2 words - "an-" , a prefix meaning 'absence of ' and directly cognate with both our English prefix 'un-' and our word 'no', and 'anta', meaning a boundary or an end and directly akin to the English word 'end' : ananta means, both etymologically and literally, 'un-end' or 'no-end' .

(Delve a bit in the etymology of modern English, and it soons appears that its closeness with ancient Sanskrit is stunning)

Nebi Caka· University of PrishtinaH Chris Ransford· Karlsruhe Institute of TechnologyManish Kumar· National Institute of Technology, HamirpurNebi Caka· University of PrishtinaH Chris Ransford· Karlsruhe Institute of TechnologyLatin comes from the common Indo European source of most European languages. I studied Sanskrit, and find it often extremely enlightening to find the common Ur-word source to modern European words . Latin or Greek or Germanic did not spontaneously arise ex nihilo but came from a much older ultimate source, common to most modern separate European families of languages, which is missed if you go much further back in time than 1 or 2 thousand years ago.

Examples ?

Candidate is ultimately the same word as hunger. Why ? The answer is fascinating.....

or : Hundred is 2 words, a first element stemming from the same Ur-source that gave the word cent, and the second element red from the selfsame ultimate source that yielded ratio: hundred is 'cent-ratio', i.e. the number one hundred.

Thousand, the 'big hundred' - the first element means big, as in thumb (the big finger), or as in tumor, or tumescence in other families of languages than Anglo-Saxon..

Weihen, to hallow or consecrate in German, is ultimately the same word as 'victim': Why?

Going back to only Germanic or only Latin or only Greek does not enlightenin the least, provides no clues to why and how the tongue developed, and explains nothing. To understand whence the words came and why, you have to go back to proto-Sanskrit and even sometimes to the Merritt's Ruhlen Nostratic. To do that is endlessly enlightening.

Modern European etymologies bear the marks of ancient wars, catastrophes, climate, geography, superstitions, religions, hunts, and so on: languages do not lie but constitute a fascinating repository of mankind's history. Again, explaining an etymology by Old English or Latin does not explain anything, and if you stop there a whole body of knowledge is not even scratched at.

Nebi Caka· University of PrishtinaI agree with you that Sanskrit word 'ananta' < ‘an+anta’ means ‘no-end’, but the English word ‘infinity’ (1350-1400) certainly doesn’t come directly from Sanskrit ‘ananta’.

H Chris Ransford· Karlsruhe Institute of TechnologyMaybe you should read posts more carefully???

Infinite, as I indicated , comes from Latin in- (cognate with un- and an-) and then a word (finis) originally meaning a wooden post stuck in the ground and marking a boundary, cognate with the Germanic word 'dike' meaning a fortified edge of water.

Not sure what your point is???

Chenyang Shen· Hong Kong Baptist UniversityH Chris Ransford· Karlsruhe Institute of Technology(Their separate infinite sizes are by the way called 'cardinality' rather than size, and 'stronger' rather than larger', because we're uneasy about speaking of the different sizes of infinity, but it's just semantics that does not change anything to the underlying reality.)

It's interesting that errors can often be routinely made when dealing with infinities. For instance, the famous 'Hilbert Hotel' scenario is often used in misleading ways: if you are going to use the usual scenario whereby a first arriving guest moves into the first (occupied) room, who moves on to the second one and so on ad infinitum, the scenario actually does not work the way it is intended to, because it takes an infinite time for the guests to occupy the hotel under that scenario: the scenario has only transformed one infinity (to wit, a second infinity of number of guests) by another infinity which pops up elsewhere ( i.e., infinite time)

Interestingly, it seems relatively easy to find an infinity even larger (stronger) than aleph(∞). Infinity is always open-ended, uncircumscribable.

Can you help by adding an answer?