Infinitely small number added infinitely?

You have only restated the principles of your previous corollary. I have already shown you that the proof of your corollary requires that "a" < "y". For an infintesimally small number added to itself an infinite number of times - cancelling out the regressive and progressive infinities, you end up with the smallest number possible - which does not have a number between itself and Zero (and yet is not equal to Zero). The proof of your corollary fails, because there can be no "a" less than this "y", i.e. the smallest number possible. The proof of your corollary does not take into account the smallest number possible, and therefore fails to prove that there is a number between the smallest number possible and Zero. The smallest number possible does not contradict your corollary. Your corollary simply does not account for a smallest possible number. The trichotomy property is not violated, as the smallest number possible > Zero. Your only arguably valid point is that there is no infintesimally small number other than zero itself. However, you still haven't told me what the sum of an infinitesimally small number added to itself an infinite number of times would equal. I say the infinities cancel out, and that's why the sum is not Zero - but the smallest number possible. I suggest you focus your attention on disproving my equation first, because it seems to me that mathematics has not contemplated my equation and therefore none of its proofs will ever take it into account. I say this because I assume that mathematicians will deny the existence of the smallest number possible. However, tell me what's wrong with the idea of adding an infinitesimally small number to itself an infinite number of times. What's wrong with the idea of canceling out the infinities? I see nothing wrong with this idea. P.S. I do appreciate your attempts to answer my questions, as I am genuinely interested in your answers. I respect constructive criticism as the tool it is for finding the truth. I am only interested in the truth, here, but am not willing to accept someone's word without understanding the logic and reasoning underlying that word. I believe I have asked a logical question, which deserves a logical refutation, concession or agreement.

 

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John J. Bannan, let's say that this number of yours that is greater than zero but has no number between it and zero is n. If this number actually exists, then what is the value of n-(n/2)? Where is that on the number line?

The problem here is that the idea of a "smallest possible number" (other than zero) is nonsensical. For any number, no matter how small, you can always just divide it by 2 and get a number that is only half the size. If there really is some number that "right next to zero" that is not zero but has no value between it and zero, then numbers are quantized. Do you have some reason to believe that numbers are quantized? No offense, but I doubt very much that you can present any evidence to support that idea.

 

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No, I am not suggesting numbers are quantized. And I understand your objection. But, you need to adequately address my hypothesis that an infinitely small number added to itself an infinite number of times would cancel out the infinities, thus creating the smallest number possible not equal to zero. I understand that current math has a big problem with that hypothesis. Nevertheless, my hypothesis is a valid question for math to deal with. You can say, "we'll an infinitely small number doesn't exist." And I can say, "we'll it does exist, because it's simply .000000 to infinity ending in 1." I don't see any mathematical reason why I can't create the number .000 . . .(infinity)1. And I don't see what's wrong with saying you can add an infinite number of this infinitely small number to itself, thus cancelling the ascending and descending infinities. Why is that not a valid mathematical question?

 

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Well 0.000 . . .(infinity)1 would be exactly equal to 0 as the one at the end of your number will never appear as the 0s go on forever.

 

Sounds like the same problem as when you say that 0.9999999999999999999999...=1, they are indeed the same number because of the infinite number of digits. Same with 0.0000...(infinitely many)1, the only way to define it is to equate it to zero. Your assumption that a smallest non-zero number exists can be shown with a little work to be in direct contradiction with the completness axiom, which is what's used to deduce the density property of real numbers. In other words, the real numbers are specifically defined so that there is no smallest positive number (note, in most conventions zero is not considered positive or negative, it falls into its own category).

Now you might ask, what if you do what mathematicians haven't considered (at least when dealing with real numbers) and construct a system in which the completness axiom doesn't apply? Well you might be able to construct a logical system of this sort, but it won't correspond to the set of real numbers. And when mathematicians deal with concepts such as infinity and the infinitesimal, it is usually in the context of real numbers, or systems built with real numbers as the foundation.

In short, you can go ahead and define your own set of axioms, which will in turn define a set of objects to which you can then attempt to give some collection of properties. But anything you prove with such a system will only be true insofar as your axioms are themselves true. I could take "Elvis is God" as an axiom if I wanted, and attempt to construct a system resembling numbers on this foundation along with some additional assumptions. Doesn't mean it relates in any way to reality or real numbers.

 

All right then. I agree with you that the current mathematic system of real numbers does not take into account an infinitely small number. But, should it? Seems to me that asking the question "what is the sum of an infinitely small number added to itself an infinite number of times" is a logical and good question to ask. Disparaging the question as saying it's like defining "Elvis=God" is unfair and ridiculous. I thought you mathematicians salivated over good math questions. Also, I can't imagine you math guys will just run and hide when a good math question is critical of a cherished axiom. If anything, I thought you would try to shoot it down using logic and reasoning.

 

Ah, but the number 1 will appear, if you cancel out the infinity by adding itself to itself an infinite number of times.

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But, don't ask me what it would look like, because I don't know. It can't be represented in the current set of mathematical symbols to my knowledge. Just for fun, call it the alpha omega number.

 

Okay, glad we agree on that.

I address it by contending that an "infinitly small number" is in fact zero, and you can add zero to itself as often as you want (infinitly many times, even) and still have zero.

This is a self-contradictory statement. If the 0.000... goes out to infinity, it cannot end in a 1. If it ever ends with anything (like a 1, for example) then by definition it does not go out to infinity.

As I said above, the problem is that your very statement of your number is self-contradictory. We cannot assume tha the 0.000... goes on forever AND that it terminates in a 1. You want to perform operations on a number with properties that no real number can have. Specifically, you want your number to be greater than zero but not have any numbers between itself and zero. This is impossible.

Again, I have to ask you: if your number is n, then where is n/2 on the number line?

 

The whole point of logic and reasoning is that you can't prove the axioms themselves. Then they wouldn't be axioms. "Elvis=God" hardly seems more ridiculous to me than "Something=Nothing" but that's just my personal opinion. The axioms of real numbers are chosen for specific reasons, to give us a logical self-consistent system which appears to have uses in the real world. As any mathematician will tell you, there is no a priori reason why the axioms of "real numbers" have to correspond to anything in reality. It's simply observed in experimental science that real numbers (and other classes of mathematical objects) seem to in fact describe the real world. Physicists often remark on this apparent and incredible coincidence, "mathematics is the language of nature".

If you want to change the axioms of real numbers to create your own system, i.e. "Bannan numbers" where there exists a smallest number greater than zero, you can go right ahead. But mathematicians won't take it seriously unless you can start deriving useful, non-contradictory results with it, and simply stating "Something=Nothing" without getting into specifics just doesn't cut it.

 

John, you are essentially talking about the hyperreal numbers (but you are doing it badly).

First some background. Both Newton and Leibniz used the concept of an infinitesimal in their development of the calculus. This concept had two basic problems. First, no matter how mathematicians tried, they could not make the concept of an infinitesimal rigorous. Second, as mathematicians added rigor elsewhere (i.e., numbers), the concept of an infinitesimal no longer even made sense. There simply is no such thing as an infinitely small number in the reals. Along the way, Weirstrass developed a rigorous development of the calculus that completely avoided the use of infinitesimals. This infinitesimal-free formulation of the calculus relies instead on the the epsilon-delta concept of a limit that trips up many freshman calculus students.

In the 1960s mathematicians did manage to make the concept of the infinitesimal rigorous by extending the real numbers. The hyperreal numbers have standard parts (i.e., the reals), infinitesimal parts, and even infinite parts. The hyperreals are a true extension to the reals, meaning that any theorem of the real numbers also pertains to the hyperreals by extension. There still is no smallest number in the hyperreals. An infinite sum of infinitesimals can be a standard number. That does not mean this is the smallest standard number.

I suggest you do some reading on the hyperreals, non-standard analysis, and non-standard calculus.

 

Well, that's a good point. If an infinitely small number is equal to zero, and you add an infinite number of them, you would end with zero. But, if you take a different order of operation, and say, an infinitely small number added to itself an infinite number of times cancels out the infinities and thus creates the smallest number possible, than you don't have zero.
Second, why can't I create an imaginary number that has an infinity of zeros to the right of the decimal point that ends in 1? You can quantify infinity - can't you? Wasn't this your argument about Zeno's paradox. That although an infinite number of fractions exists between, let's say, 1 and 0, that the infinite set still sums up to 1? Isn't 1 a quantity? Why can't I perform a similar trick with an infinite number of zeros, and claim there is an amount to which that infinity adds up which is exactly cancelled out by a similar amount of zeros being added on? And then, put a 1 on the end?

 

The ellipsis in 0.000... means that the zeros go on forever. There is no end to the sequence. There is nowhere you can append your 1. 0.000...1 makes just about as much sense as \(1+2*\). Both are ill-formed expressions.

 

Actually, "Elvis=God" is far more ridiculous than "Nothing=Something." Science can't explain why anything even exists, and unfortunately for us, the only candidate for the job appears to be nothingness. Remember, the Big Bang suddenly springs from nowhere? Where did the Big Bang come from, if from nowhere? Now, sure, maybe there is something else out there which we haven't yet encountered that could explain this. But, to say "nothingness=somethingness" is silly in light of our current model of the Big Bang springing from nothingness - is itself silly.

 

I can, if I cancel the infinities out.

 

Why would the infinities cancel out?
If you take 0.0recurring as equal to 0 there is no infinite decimal to cancel anyway.
The 1 at the end does not exist.

 

Hey, sounds good.

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But why would you assume that I meant the sum of an infinitely small number added to itself an infinite number of times to equal a standard number? You clearly won't find this number on the number line. Is this number a hyperreal number?

 

Simple. Imagine a 1 at the end of an infinite number of zeros right of the decimal point. Now, imagine an infinite number of these 1s being added to themselves pushing the infinity of zeros in the opposite direction toward 1. The constant state of addition would exactly counterbalance the constant state of infinite regression - creating the smallest number possible. But, perhaps it is better to describe it as the smallest hyperreal number possible.

 

There is no end to infinity
Infinity - 1 = infinity
Infinity + 1 = infinity
So no matter what you try and do there is still going to be infinite 0s before the non existent 1

 

Again, science doesn't attempt (as of yet) to explain where the Big Bang came from. If time started at the Big Bang, then there's no such thing as "before" the Big Bang, because time needs to exist in order for there to be a "before". Or time could have existed before the Big Bang, and gone infinitely far into the past. Nothing in physics says that causality is a fundamental property of existence- that's an assumption you make. We physicists and mathematicians simply shrug our shoulders when asked these kinds of questions, rather than attempting to construct an explanation which could potentially be incorrect.

If you think you have an answer to this problem, then you need to show how your assumptions can be used to derive what we observe in the universe, as well as showing why there are things that we don't observe. Otherwise, your speculation is useless from a scientific point of view. You may think "Something=Nothing" isn't a ridiculous assumption, but that's a matter of personal opinion, and logic itself is independent of personal opinion or common sense.

 

You forgot, infinity-infinity, which apparently is undefined. See http://mathforum.org/library/drmath/view/57069.html