Approaching infinity

Jon

Jon

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#1
#1
Following on from another thread involving infinity, I had a new insight. Or is it a question of sorts?

Is the number 1,000,000,000,000,000,000,000,000 closer to infinity than the number 5? I am sure you will agree that both numbers are infinitely far away from infinity. If so, they are an equal distance from infinity, because an infinite distance is not larger than an infinite distance. A paradox.

Should this be in the Approaching infinity thread or the Paradox thread? It is itself a paradox.
 
T

The_Doc_Man

Founding Member
#2
#2
Not a paradox, but rather an example of Goedel's Theorem at work. Basically, a language can be correct or complete but can't be both.

In English, you can talk about "approaching infinity" as though such a thing could happen. The scientifically correct description is that something can increase or be extended "without limit." Science describes things as having "converging behavior" or "non-converging behavior." A numerical series based on larger and larger values will either converge/approach some limit or it will not.

In the paradox thread, you brought up a paradox and I offered another. In both cases, the implied series was "converging" in nature and it would be correct to say that the series "approached X as n increased."

In scientific terms, NO definite number is closer to infinity than any other. If you can write a number of 100,000 digits, it is no closer to infinity than a number written with 1 digit - because (as you point out) an infinite number of distinct digit sequences, each member greater than your referenced number, can be created/defined. But you cannot say they are an "equal distance" from infinity - because math doesn't let you compare infinities of the same order. Both of those numbers are from the same series - the series of enumerable numbers. That comparison is disallowed - because math is correct. But you can say it in English because English is complete.

The distinction is that in math I cannot legally write a statement equivalent to the English sentence: "This statement is false." The issue is that formal math does not allow self-meta-references to statement truth values. Only spoken languages do that.
 
Jon

Jon

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#3
#3
In essence, both perspectives are true, depending on your lens. Although we do differ a little on the detail.

Perhaps mathematically they are equally distant, given the strict constraints of our mathematical paradigm. Or is it a case that an infinite number is so far beyond the realm of understanding in size, that both numbers I quoted converge on being infinitesimally small, by comparison. Like a pin prick compared to the size of the universe. The difference being, that the analogy I used was still finite. Bring in the infinite and the two numbers I quoted converge to zero, even if they are not zero. Yes, that sounds odd to say, just as it sounds odd to say that 0.9 recurring in fact equals 1.
 
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The_Doc_Man

Founding Member
#5
#5
Yes, conceptually. No, technically, since infinity is a number for which ordinary math operators really don't apply.

In the Access World forum, we have a similar issue having to do with properties in the scope of set theory. Null - 1 is still null. So is null + 1 and null times 2 and null divided by 2. One could argue (as one limiting viewpoint) that the normal math operators apply only to normal numeric quantities. If you claim that infinity is a numeric quantity, the question remains as to whether it is a NORMAL numeric quantity. And some folks would say "NO" - which would imply that to perform ANY ordinary mathematical operation that involves infinity is actually a violation of the rules of math.
 
Jon

Jon

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#6
#6
My understanding is that maths has its own language. And in that language, 0.9 recurring means 0.99999 to infinity. It is not a semantic fudge, it is a clear way of describing that the 9's don't stop. So in the language of maths, "recurring" = "infinitely recurring". If it doesn't mean that, when does the recurring stop? If it doesn't, surely that is infinite?

If you were wondering where I am going with this, my argument is...

infinity - 1 = infinity = infinity + 1

Therefore, -1 = 0 = +1

Q.E.D.

:ROFLMAO::ROFLMAO::ROFLMAO::ROFLMAO:
 
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The_Doc_Man

Founding Member
#7
#7
Ah, but infinity is not ONLY inclusive of 0 (as in, there are an infinite number of integers) - but it is SIMULTANEOUSLY inclusive of zero and EVERY OTHER number AT THE SAME TIME.

Since infinity is not a scalar number, but rather is an ambiguous concept that has a (literally infinite) number of possible members in a set, use of the mathematics operators is not technically valid. Your proof fails on the violation of using the math operators on a number that is outside of their intended scope.

You can do anything mathematical you want to infinity and it is still infinity. The math operators have NO EFFECT on it. You need special cases for those situations where you CAN affect infinity. For instance, there are multiplicative situations where you can reach aleph-1 (where aleph-0 is the ordinary, run-of-the-mill infinity), aleph-2, etc. There are "higher levels" or perhaps "higher orders" of infinities.

When it comes to the standard math operators, infinity is generally outside of their scope of influence. So for THAT reason, "infinity - 1 = infinity." BUT your proof fails because the math symbols have no effect on infinity - but they DO take effect on ordinary numbers. So in USA football, the yellow penalty flag would come out for "Illegal shift" or "Illegal formation."

You question about "recurring" progressions is in alignment with limit theorems where a progression is taken to its extreme. (Egad! It has been literally FIFTY YEARS since I dealt with this stuff...). If you take this progression (and sorry, but I'm not going graphics on this:)

X = Sum (for I ranging from 1 to infinity) of ( 9 divided by (ten to the Ith power) )

The first few steps are 0.9, 0.99, 0.999, 0.9999, (that's the first four, you have an infinite number of steps remaining). The LIMIT of that series is 1, though mathematically, it will never actually reach that value. Another variant of Zeno's paradox, more or less.
 
Jon

Jon

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#8
#8
I think I recall a conversation about this a long long time ago, and talk that infinity came under complex numbers or something like that. There is a whole sphere of maths that defies common logic, going under different rules.

I am aware that logically -1 does not equal 0 or 1. So by deduction, you could suggest that the operators of = or + or - are at fault. To get there without prior knowledge is hard, since the assumption is that they are logical and usable under all conditions. Thinking outside of the box lets you zoom out to factor in the normally unfactorable. It takes flexibility of mind to escape the channeled thought. Switching lanes when we have been told to stay in lane is a skill within itself.

I used to love reading Edward de Bono's stuff. Lateral thinking et all. He used to do these books with symbols, circles overlapping and so on. It appealed to my logical mind, yet the lazy side of me meant I never finished any of his books!
 
T

The_Doc_Man

Founding Member
#9
#9
Not familiar with de Bono, but I have read (and thoroughly enjoyed, and learned a lot from) The Eternal Golden Braid by Douglas Hofstadter (think I got the name right). It is a book about perspectives told from three different viewpoints besides his own: Goedel, Escher, and Bach. Their work was heavily involved in something that a really good system analyst needs to understand - that a fixed perspective NEVER sees everything. You need that flexibility to mentally snap between reductionism (seeing the details) and wholism (seeing the system in which those details somehow fit).

Turns out you need that ability in some parts of physics, too, because there were a LOT of proofs about individual vs. group properties. Some folks proved the old Gas Law: PV = nRT. They did so by considering the transition from seeing an enclosed gas as a bunch of molecules moving randomly AND a mass of gas-phase material for which properties of the group (i.e. the molecules taken as a whole) can be described. Turns out there are other cases like that, too. Superconductivity can be discussed based on electron domains and a phenomenon called "delocalization" (related to the idea of "distant interaction" and "quantum entanglement"). Magnetism works based on atomic domains and their power as a group (well, as a crystalline metallic material), too.
 
Jon

Jon

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#10
#10
That reminds me of the saying, "If all you have is a hammer, everything looks like a nail." The same can be said of Quantum Physics. Scale matters.

[Edit: If the above is not clear, I am referring to fixed perspective.]
 
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Jon

Jon

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#12
#12
That is why I was arguing the toss about the "debunked" Jungarian perspective. There is always another point of view.
 
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