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What is infinity?
Posted: 29 January 2010 12:06 PM   [ Ignore ]
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The concept of infinity and what it is has baffled philosophers for more than two thousand years.

Here are some definitions of infinity:

1. Unlimited extent of time, space, or quantity; eternity; boundlessness; immensity.

2. Endless or indefinite number.

3. A quantity greater than any assignable quantity of the same kind.

Intuitively, we think of infinity as something or entity so humongous (or so minute) that its size or numerical value cannot be determined at all.

Imagine the number of viruses (the smallest known biological entities) on the earth, the number of quantum “particles” in the universe, the amount of matter and energy (including dark) or the size of the expanding universe itself. In other words, infinity lurks everywhere if one looks carefully.

Mathematicians, physicists and cosmologists have developed techniques to deal with extremely large or small quantities. However, these cannot determine what is infinite because if so, then the final value determined is not infinite. This is the conundrum or paradox of infinity.

In the 19th Century, Georg Cantor, using set theory (which he created), proposed a revolutionary approach to define mathematical infinity.

From the wiki on infinity

A set of elements can be defined as infinite if the set has a seemingly paradoxical quality: a subset of elements in an infinite set can be matched up, one-to-one, to all of the elements in a set.

Consider the following:

1,2,3,4,5….......(the set of natural numbers) (NN)
1.3.5.7,9….......(the set of odd numbers) (ON)

There is a one-to-one correspondence for every number in both sets (if the number of numbers in both sets are assumed to be inexhaustible), therefore NN=ON and both sets are infinite.

Mathematically brilliant, except for the oddness of the outcome which arises from the definition of an infinite set whereby a subset (ON) is equal to the whole set (NN).

The part is equal to the whole contradicts Euclid’s Principle: “The whole is greater than the part”.

The other objection to Cantor’s approach is the reliance on induction. Unlike a finite set, an infinite set has undefined number of elements and it cannot be assumed there will be one-to-one correspondence for every element of both sets based on the experience of finite counting.

The next objection comes from finitism

In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists.

Another approach to inquire into the nature of infinity is to define vagueness as the essence of infinity. In the quantum realm, quantum objects are vague objects and empty space is not empty, it has “vacuum energy” and out of this infinite sea of vague reality, matter and energy emerges. Cosmologists theorize that after the big bang, space expanded exponentially (inflation) faster than the speed of light and is still expanding in the observable universe, but the actual universe is much larger than the observable universe.

William Blake’s sublime opening stanza in “Auguries of Innocence” aptly describes finding infinity:

To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.

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Posted: 29 January 2010 12:46 PM   [ Ignore ]   [ # 1 ]
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It never made sense to me to think of infinity as a number, but rather as a direction.

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Posted: 29 January 2010 02:15 PM   [ Ignore ]   [ # 2 ]
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Infinity is reality undivided. It is represented by the number 1 and contained in the symbol 0. 0 reflects nothing, but the abstract notion of it makes 1. Reality as it is, transcends conceptual distinction. The identity of ‘something’ is the marvelous ability of Maya, without which we would not experience anything at all. Maya is self-love. The reflection of an Image of the infinite. Infinity is devoid of identity.

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Posted: 29 January 2010 06:31 PM   [ Ignore ]   [ # 3 ]
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As I seem to recall, there are different orders of infinity.  For example,  ∞ * 2 = ∞,  but ∞ * ∞ = aleph (a different order of infinity).  So, the infinity of all numbers is just twice the infinity of all odd numbers, therefore it’s the same infinity. 

And mentioning all the viruses or all the quanta of the universe are certainly extremely large numbers, but by definition are not even close to infinity.

Occam

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Posted: 29 January 2010 06:43 PM   [ Ignore ]   [ # 4 ]
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Occam - 29 January 2010 06:31 PM

As I seem to recall, there are different orders of infinity.  For example,  ∞ * 2 = ∞,  but ∞ * ∞ = aleph (a different order of infinity).  So, the infinity of all numbers is just twice the infinity of all odd numbers, therefore it’s the same infinity. 

Not sure about ∞ * ∞; that might be ill-defined, since really the symbol “∞” doesn’t distinguish between these different orders. But IIRC the lowest order of infinity is the infinity of the integers (1,2,3,4 ...). The infinity of the real number line is a larger order of infinity, because for every integer (e.g., the number “1”) there are an infinity of real numbers. (e.g., 1.08342342, pi, e, etc.)

The mathematics of infinities is different from the mathematics of integers or reals; that doesn’t mean (as some seem to think) that therefore the mathematics of infinities is ill-defined. It’s just different. Infinity plus one is still infinity. Infinity times three is still infinity, as you say.

Occam - 29 January 2010 06:31 PM

And mentioning all the viruses or all the quanta of the universe are certainly extremely large numbers, but by definition are not even close to infinity.

Indeed. In fact, they are just as far from infinity as is the number one. That is, they are infinitely far.

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Posted: 29 January 2010 08:20 PM   [ Ignore ]   [ # 5 ]
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KurtJ - 29 January 2010 12:46 PM

It never made sense to me to think of infinity as a number, but rather as a direction.

Quite so, however conceiving infinity as a direction brings up the question of a direction to where and to which direction?

Do you mean Directed Infinity?

A directed infinity in direction z is an infinite numerical quantity that is a positive real multiple of the complex number z.

A directed infinity with unknown direction is known as complex infinity.

Please explain.

[ Edited: 29 January 2010 08:23 PM by kkwan ]
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Posted: 29 January 2010 09:31 PM   [ Ignore ]   [ # 6 ]
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Occam - 29 January 2010 06:31 PM

As I seem to recall, there are different orders of infinity.  For example,  ∞ * 2 = ∞,  but ∞ * ∞ = aleph (a different order of infinity).  So, the infinity of all numbers is just twice the infinity of all odd numbers, therefore it’s the same infinity. 

And mentioning all the viruses or all the quanta of the universe are certainly extremely large numbers, but by definition are not even close to infinity.

Occam

From the wiki on Aleph number

In set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph.

The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

This goes on and on and it implies there is an infinity of infinities which to Cantor, who was a very religious man, was God.

There is Controversy over Cantor’s theory

At the start, Cantor’s Theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: “I don’t know what predominates in Cantor’s theory - philosophy or theology, but I am sure that there is no mathematics there.” Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Cantor’s ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: “No one will drive us from the paradise which Cantor created for us” (Hilbert, 1926). To which Wittgenstein replied “if one person can see it as a paradise of mathematicians, why should not another see it as a joke?”

As regards viruses, it should be noted that they are not static in number (they multiply, or die off profusely), quantum “particles” emerge and annihilate spontaneously, there is no definite value and no end in their numbers and therefore it is not possible to determine the numerical value of the both of them.

Thus, from my 2nd definition of infinity: endless or indefinite number, they can be considered as infinite.

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Posted: 29 January 2010 11:21 PM   [ Ignore ]   [ # 7 ]
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dougsmith - 29 January 2010 06:43 PM

The mathematics of infinities is different from the mathematics of integers or reals; that doesn’t mean (as some seem to think) that therefore the mathematics of infinities is ill-defined. It’s just different. Infinity plus one is still infinity. Infinity times three is still infinity, as you say.

There is Controvery over Cantor’s theory

At the start, Cantor’s Theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: “I don’t know what predominates in Cantor’s theory - philosophy or theology, but I am sure that there is no mathematics there.” Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Wittgenstein denies Hume’s principle, arguing that our concept of number depends essentially on counting. “Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one”.

A common objection to Cantor’s theory of infinite number involves the axiom of infinity. It is generally recognised view by all logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury (1979, p. 305) has argued “there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite”. Bertrand Russell for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry (2000, p. 10) has noted that “The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor’s axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all”.

Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl wrote:

“    … classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor’s] set theory ….”

Richard Arthur, philosopher and expert on Leibniz, has argued that Cantor’s appeal to the idea of an actual infinite (formally captured by the axiom of infinity) is philosophically unjustified.

Cantor, who was a very religious man, went on to propose an Absolute Infinite which to him, was God.

The Absolute Infinite is mathematician Georg Cantor’s concept of an “infinity” that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.

It should also be noted that in the argument for EN=ON=NN using Cantor’s definition of an infinite set, we are misled to an absurd conclusion by uncritically applying Hume’s principle:

1,2,3,4,5…......Natural numbers (NN)
1,3,5,7,9…......Odd Numbers (ON)

If the members of both the sets are indistinguishable, then Hume’s principle (one-to-one correspondence) can be used to determine the cardinality of the sets. Not so here, and by matching identical numbers (except for 1), it is obvious that 2,4,6…..even numbers in (NN) cannot be matched by identical numbers from the set of odd numbers (ON). In other words, there are members of NN which are not in ON. Therefore, there is no logical basis to change the method of counting in this instance (notwithstanding Hume’s Principle) for any value, including infinity.

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Posted: 30 January 2010 01:32 AM   [ Ignore ]   [ # 8 ]
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I have no clue about infinity other than that it is infinite.
However just for clarifying my thinking:

a) 1+1+1+1+1+1 = 6
b) 2 + 2 + 2 = 6
c) 3 + 3 = 6

Does the result in a) contain more numbers than the result in b) or c)?

If it does not, how can one argue that one infinity has more numbers than another infinity?
If it does, is there a result named infinity (+1) or (+2), etc?
Or can infinity ever be a result at all?

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Posted: 30 January 2010 04:24 AM   [ Ignore ]   [ # 9 ]
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kkwan - 29 January 2010 11:21 PM

There is Controvery over Cantor’s theory

There is controversy over everything. Cantor’s theory is well established in mathematics, as it says quite plainly in your wiki article: ” ... this work has found near-universal acceptance in the mathematics community”.

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Posted: 30 January 2010 10:18 AM   [ Ignore ]   [ # 10 ]
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dougsmith - 30 January 2010 04:24 AM
kkwan - 29 January 2010 11:21 PM

There is Controvery over Cantor’s theory

There is controversy over everything. Cantor’s theory is well established in mathematics, as it says quite plainly in your wiki article: ” ... this work has found near-universal acceptance in the mathematics community”.

It does, with the superscript (citation needed) in the wiki, and:

“it has been criticized in several areas by mathematicians and philosophers.”

From this thought provoking article on
Wittgenstein’s Philosophy of Mathematics

Closely related with this conflation of intensions and extensions is the fact that we mistakenly act as if the word ‘infinite’ is a “number word,” because in ordinary discourse we answer the question “how many?” with both. But “infinite’ is not a quantity,” Wittgenstein insists; the word ‘infinite’ and a number word like ‘five’ do not have the same syntax. The words ‘finite’ and ‘infinite’ do not function as adjectives on the words ‘class’ or ‘set,’, for the terms “finite class” and “infinite class” use ‘class’ in completely different ways. An infinite class is a recursive rule or “an induction,” whereas the symbol for a finite class is a list or extension. It is because an induction has much in common with the multiplicity of a finite class that we erroneously call it an infinite class

Why Set Theory is wrong:

“Set theory is wrong” and nonsensical , says Wittgenstein, because it presupposes a fictitious symbolism of infinite signs instead of an actual symbolism with finite signs. The grand intimation of set theory, which begins with “Dirichlet’s concept of a function”, is that we can in principle represent an infinite set by an enumeration, but because of human or physical limitations, we will instead describe it intensionally.  But, says Wittgenstein, “there can’t be possibility and actuality in mathematics,” for mathematics is an actual calculus, which “is concerned only with the signs with which it actually operates”. As Wittgenstein puts it at, the fact that “we can’t describe mathematics, we can only do it” in and “of itself abolishes every ‘set theory’.”

Perhaps the best example of this phenomenon is Dedekind, who in giving his ‘definition of an “infinite class” as “a class which is similar to a proper subclass of itself”, “tried to describe an infinite class”. If, however, we try to apply this ‘definition’ to a particular class in order to ascertain whether it is finite or infinite, the attempt is ‘laughable’ if we apply it to a finite class, such as “a certain row of trees,” and it is ‘nonsense’ if we apply it to “an infinite class,” for we cannot even attempt “to co-ordinate it” , because “the relation m = 2n [does not] correlate the class of all numbers with one of its subclasses”, it is an “infinite process” which “correlates any arbitrary number with another.”  So, although we can use m = 2n on the rule for generating the naturals (i.e., our domain) and thereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doing so we do not correlate two infinite sets or extensions.  If we try to apply Dedekind’s definition as a criterion for determining whether a given set is infinite by establishing a 1-1 correspondence between two inductive rules for generating “infinite extensions,” one of which is an “extensional subset” of the other, we can’t possibly learn anything we didn’t already know when we applied the ‘criterion’ to two inductive rules. If Dedekind or anyone else insists on calling an inductive rule an “infinite set,” he and we must still mark the categorical difference between such a set and a finite set with a determinate, finite cardinality.

Indeed, on Wittgenstein’s account, the failure to properly distinguish mathematical extensions and intensions is the root cause of the mistaken interpretation of Cantor’s diagonal proof as a proof of the existence of infinite sets of lesser and greater cardinality.

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Posted: 30 January 2010 11:40 AM   [ Ignore ]   [ # 11 ]
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Cantor’s theorem that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostilitythan any other mathematical argument, before or since. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this “harmless little argument”, asking, “what had it done to anyone to make them angry with it?”

From you Wikipedia article.

Why this crusade against infinity? It really is accepted by nearly all mathematicians! I wonder why you are struggling for the second time on this forum with different kinds of infinity. (No citations or links please. I do not read them, I promise.)

GdB

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Posted: 30 January 2010 11:45 AM   [ Ignore ]   [ # 12 ]
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Write4U - 30 January 2010 01:32 AM

I have no clue about infinity other than that it is infinite.
However just for clarifying my thinking:

a) 1+1+1+1+1+1 = 6
b) 2 + 2 + 2 = 6
c) 3 + 3 = 6

Does the result in a) contain more numbers than the result in b) or c)?

If it does not, how can one argue that one infinity has more numbers than another infinity?
If it does, is there a result named infinity (+1) or (+2), etc?
Or can infinity ever be a result at all?


All the results are 6. No.
There are more 1s than 2s or 3s.

How does that relate to infinity?
Please explain.

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Posted: 30 January 2010 12:24 PM   [ Ignore ]   [ # 13 ]
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GdB - 30 January 2010 11:40 AM

Cantor’s theorem that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostilitythan any other mathematical argument, before or since. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this “harmless little argument”, asking, “what had it done to anyone to make them angry with it?”

From you Wikipedia article.

Why this crusade against infinity? It really is accepted by nearly all mathematicians! I wonder why you are struggling for the second time on this forum with different kinds of infinity. (No citations or links please. I do not read them, I promise.)

GdB

Cantor’s “harmless little argument”, if it is accepted uncritically, merrily leads one to accepting a hierarchy of infinities culminating in an “infinity of infinites” which to him, was God.

This “harmless little argument” is theology masquerading as mathematical insight.

There is no crusade against infinity. Cantor’s argument, however, is not “accepted by nearly all mathematicians” because its implications are outrageous.

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Posted: 30 January 2010 12:57 PM   [ Ignore ]   [ # 14 ]
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kkwan - 30 January 2010 11:45 AM
Write4U - 30 January 2010 01:32 AM

I have no clue about infinity other than that it is infinite.
However just for clarifying my thinking:

a) 1+1+1+1+1+1 = 6
b) 2 + 2 + 2 = 6
c) 3 + 3 = 6

Does the result in a) contain more numbers than the result in b) or c)?

If it does not, how can one argue that one infinity has more numbers than another infinity?
If it does, is there a result named infinity (+1) or (+2), etc?
Or can infinity ever be a result at all?

All the results are 6. No.
There are more 1s than 2s or 3s.
How does that relate to infinity?
Please explain.

So a), b), c) are sets (quantities) of numbers which yield the same result?  I seem to recall that someone mentioned that infinity can never yield a result because infinity is always at least 1 greater than any result. So how can a set of infinite amount of numbers be larger than any other set of of infinite amount of numbers if infinity is always greater than the set? How can you close the set for counting the numbers if the result is always less than infinity? It sounds to me a little like asking are there more apples in an infinite set of apples as there are pears in an infinite set of pears? I guess one could ask if in an infinite number of fruits, there is a greater representation of apples than pears. But even that sounds incomplete to me, because if we try to count the actual number of apples and pears, each number would still be infinitely large.

I have never contemplated infinity as something that could be quantified, so please forgive me if I am trying to get a rudimentary understanding of the concept of a quantifiable infinity in mathematics..

[ Edited: 30 January 2010 02:26 PM by Write4U ]
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Posted: 30 January 2010 01:59 PM   [ Ignore ]   [ # 15 ]
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kkwan - 30 January 2010 12:24 PM

Cantor’s “harmless little argument”, if it is accepted uncritically, merrily leads one to accepting a hierarchy of infinities culminating in an “infinity of infinites” which to him, was God.

This “harmless little argument” is theology masquerading as mathematical insight.

There is no crusade against infinity. Cantor’s argument, however, is not “accepted by nearly all mathematicians” because its implications are outrageous.

Firstly, I recall hearing a proof that there is no largest infinity, e.g., that encompasses all the other infinities. Hence there is no such thing as the infinity of infinities, as though it were a thing of defined size and character. If this recollection is correct, then there is nothing for God to be, since there is no such thing as “the infinity of infinities”.

Secondly, even if there were such a thing as “the infinity of infinities”, it would not be God. God is a person who responds to prayer and created the world. An abstractum like a number or an infinity is none of those things.

Thirdly, although I have not personally polled all mathematicians, my sometime dealings with them leave me quite confident that the vast majority, indeed “nearly all”, accept Cantor’s argument. It is taught as absolutely standard mathematics.

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