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But what does .9999999999… MEAN?

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Not to rehash what had been done in the other thread, but Mindy,

What does "infinitely close" and "infinitesimally small" mean? Also, how can any reasonable concept associated to something labeled 0.999... have a right-most digit? While we're at it, why doesn't your argument about the "acceptable" infinite series formed by repeatedly halving things "to infinity" and re-summing not work for 0.999...? After all, you're just repeatedly taking a tenth of a unit into infinity and adding it back together.

I can peform the operations you say can't be done to 0.999... quite well, so I don't see why you say it isn't a number.

See Aleph_0's post #10. I meant "arbitrarily small/close."

Right, it doesn't have a right-most digit, so how do you multiply it? When you multiplied .999... by 2, what did you get?

Do you mean infinitely dividing a number by 10? Because that is not the same as dividing once and getting a non-terminating quotient.

-- Mindy

Edited by Mindy
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See Aleph_0's post #10. I meant "arbitrarily small/close."

Oh good-- otherwise I'd have to make fun of you for having the last name Newton and arguing for infinitesimal quantities. Ghosts of departed quantities, I say!

Right, it doesn't have a right-most digit, so how do you multiply it? When you multiplied .999... by 2, what did you get?

I regard real numbers as (equivalence classes of ) Cauchy sequences. To show that you get 2 as the answer, take the representative (2, 2, 2, ...) for 2 and (0.9, 0.99, 0.999, ...) for .999... . Multiplication is termwise, so the product is the sequence (1.8, 1.98, 1.998, ...), the terms of which (as you note) become as close to 2 as you like, provided you're willing to go far enough out. So each individual term *does* have an 8 at the end, if you like, but the limit is still 2.

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Do you mean infinitely dividing a number by 10? Because that is not the same as dividing once and getting a non-terminating quotient.

What it sounded like you were taking about was a "Zeno's paradox" process of allowing 1 = 1/2 + 1/4 + 1/8 + ... . If so, this is the exact same problem as .999... = 1-- in fact, in binary this fact is rendered as 1 = 0.1111..... . If that's not what you meant, sorry for misunderstanding.

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As discussed earlier (and in the other thread), "0.999..." refers to the following:

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What I see as a reasonable objection to the above concept of method is the invocation of an infinite process, but this is not necessary. A common description of the above limit might be, "The value of the sum of the sequence if it were to be carried out to inifinity." I would agree this is a bad description, because no sequence can actually be carried out inifinitely. However, you can describe the evaluation of the above limit in a well-defined way which corresponds to reality, as such:

The evaluation of the limit is equal to the smallest real number which is greater than the sum of the sequence after n iterations, where n may be any natural number greater than or equal to 1.

The value of the limit does not have to be described as something which is approached or as the evaluation of an impossibly infinite sequence. It is mathematically well-defined and conceptually valid (i.e. connected to reality).

Edit: replace "is" with "may be"

Edited by Jake
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Two responses to this: One, I have no problem with a convention that substitutes a sequence's limit with its "numerical value," since it is merely a matter of convenience. I just note that that limit was not calculated to be equal, it was calculated to be the limit. The mathematical part leaves them unequal, the conventional part sets them equal.

My other point is the thought that if you resort to treating the problem of multiplying .999... by 2 in such an elaborate way as equivalence classes of Cauchy sequences, aren't you implying that .999... is not, in the direct mathematical sense, equal to 1?

I must be missing something--how are they the exact same problem?

-- Mindy

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It seems that the concept of method claims that in mathematics we are still dealing with quantities of real-world things, and that's how my claim differs from it. I would agree that, at introductory stages of language education, when we speak of natural numbers, we are in some sense referring to a quantity of things. However, that's not what we do in mathematics at any stage. For instance, if it turned out that the universe were finite--as far as I know, there is no reason to suspect that it's finite or infinite in, say, the number of electrons that exist--that would constrain our initial notion of counting because there would be some very large quantity, beyond which no quantity would successfully refer.

Arithmetic would remain oblivious to this fact, since arithmetic assumes an infinity of natural numbers, and so within the system of arithmetic, every number is meaningful and no number attempts to refer at all. Instead, numbers are meant to represent. At the level of arithmetic, this representation is so near to what it attempts to represent, that it's hard to see the difference between the representation and the thing represented. But in arithmetic, we simply define a primitive element, which we call 0, and then define a function called the successor function, and define our domain as the closure of 0 and the successor function. From there we define notions of addition, multiplication, and limited subtraction and division. This is not meant to refer to the quantity of apples in some basket, or any quantity at all, though this is meant to behave in a way that mirrors counting--so that you may use this system in order to do your counting for you. The reason for having such a system is because, even if there is a largest quantity, we may never learn what it is or even care what it is. It is more desirable to simply have a system which is agnostic about there being a largest quantity, and which is effective under any hypothesis.

So in the end, arithmetic is nothing more than a linguistic practice which allows us to describe the world. When we group things into fives, count the groups, and multiply to obtain the quantity of things, we are using an abstract mathematical construct which behaves the same way that counting does. How we know that the behavior is the same is hard to spell out, but seems beside the point for this particular conversation.

The initial sense is one of a pragmatic approach. As the concept of method is the 'how' we expand the numbers, and in continuous quantity, it permits the 'how' of refinement of resolution. This is done by setting aside for the moment the 'what' is being quantified or is being used as the unit.

The reason to suspect that the universe is finite in the context of quantity of electrons would be the law of identity. Treating arithmetic as oblivious, or as being agnostic seems to suggest a severing of the logical hierarchy between arithmetic and what perceptually gives rise to its necessity.

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Two responses to this: One, I have no problem with a convention that substitutes a sequence's limit with its "numerical value," since it is merely a matter of convenience. I just note that that limit was not calculated to be equal, it was calculated to be the limit. The mathematical part leaves them unequal, the conventional part sets them equal.

My other point is the thought that if you resort to treating the problem of multiplying .999... by 2 in such an elaborate way as equivalence classes of Cauchy sequences, aren't you implying that .999... is not, in the direct mathematical sense, equal to 1?

That's right-- strictly speaking I should have said the result of multiplying (2, 2, ...) and (0.9, 0.99, ...) is *equivalent to* (2, 2, ...) again. This is why a real number is the *set of equivalence classes* of such sequences, not just one of them. I can see how that part would make people awfully jumpy, since there are necessarily infinitely many such sequences for any real number. So while you're certainly right that the two *representatives* of the real number commonly referred to as (1, 1, ...) and (0.9, 0.99, ...) are distinct sequences, they equal the same number. Regarding equivalent sequences as the same real number is to avoid introducing quantities smaller than any rational number, which'd just be useless, since as we all know here A is A and a unit of measuring length has a definite length and hence an error.

I must be missing something--how are they the exact same problem?

I was referring to this line:

"There are infinite series that do, in fact, add to a finite number. Cut a quantity in half, repeatedly into infinity, then add it all back up, and the infinite series equals the original quantity, of course. But non-terminating decimals are not in this category."

Why not instead say:

"There are infinite series that do, in fact, add to a finite number. Cut a quantity in ten pieces, take one of them, cut it into ten pieces, take one of *those*, repeatedly into infinity, then add it all back up, and the infinite series equals the original quantity, of course. But non-terminating decimals are not in this category."

Because that's exactly what 0.999... is getting at. I don't see why you're allowing yourself to half something indefinitely, but not cut into tenths.

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An integer number is at the same time a real number. In a purely mathematical context, 1.0 is the same object as 1. There are contexts in which they are interpreted somewhat differently, for example as results of a length measurement, where the number of decimals conveys some information about the precision of the measurement, even if the decimals are all 0s. Also, in computer programming, internally some compilers treat 1 and 1.0 differently.

Sasha

An entirely satisfactory answer, thank you. Those other contexts you mentioned are my primary contexts in working with numbers.

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I was referring to this line:

"There are infinite series that do, in fact, add to a finite number. Cut a quantity in half, repeatedly into infinity, then add it all back up, and the infinite series equals the original quantity, of course. But non-terminating decimals are not in this category."

Why not instead say:

"There are infinite series that do, in fact, add to a finite number. Cut a quantity in ten pieces, take one of them, cut it into ten pieces, take one of *those*, repeatedly into infinity, then add it all back up, and the infinite series equals the original quantity, of course. But non-terminating decimals are not in this category."

Because that's exactly what 0.999... is getting at. I don't see why you're allowing yourself to half something indefinitely, but not cut into tenths.

Not all infinite series sum to a finite number. That was my point. By creating the series in a simple way, I could show easily that such a sequence could add up as required. That it was created by division was merely a matter of convenience. If we took .999... and divided it infinitely by 10, then added it all back up, we'd get .999... That doesn't get us anywhere in regards to .999... = 1.

-- Mindy

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Not all infinite series sum to a finite number. That was my point. By creating the series in a simple way, I could show easily that such a sequence could add up as required. That it was created by division was merely a matter of convenience. If we took .999... and divided it infinitely by 10, then added it all back up, we'd get .999... That doesn't get us anywhere in regards to .999... = 1.

-- Mindy

So you were just arguing that the infinite series you alluded to with the halving process converges to a finite sum, not that that sum ought to be equal to 1. Out of curiosity, what is your opinion as to the sum of 1/2 + 1/4 + 1/8 + ...?

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So you were just arguing that the infinite series you alluded to with the halving process converges to a finite sum, not that that sum ought to be equal to 1. Out of curiosity, what is your opinion as to the sum of 1/2 + 1/4 + 1/8 + ...?

It adds up to 1. Just like .999 + .000999... adds up to .999...

The mathematical incommensurability of the diameter and circumference of a circle depends on the fact that the non-terminating fraction relating those two measurements is non-terminating.

Help me with a confusion, if you will. We have a sequence, (.9, .99, .999, ... ) and the "biggest value" of that sequence, the "limiting value" is .999... . Here's my confusion. Is this limiting value a member of that sequence? If not, why not? If it is, then it is not itself the limit of the sequence. If it is not, isn't it the limit of the sequence?

I seem to be reading that there is a sequence, it has a limit, and that limit has a limit. A sequence with a limiting value and a limit, ok, because they are both defined by the sequence. I don't know how a limit can be said to have a limit...

-- Mindy

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It adds up to 1. Just like .999 + .000999... adds up to .999...

The mathematical incommensurability of the diameter and circumference of a circle depends on the fact that the non-terminating fraction relating those two measurements is non-terminating.

Help me with a confusion, if you will. We have a sequence, (.9, .99, .999, ... ) and the "biggest value" of that sequence, the "limiting value" is .999... . Here's my confusion. Is this limiting value a member of that sequence? If not, why not? If it is, then it is not itself the limit of the sequence. If it is not, isn't it the limit of the sequence?

I seem to be reading that there is a sequence, it has a limit, and that limit has a limit. A sequence with a limiting value and a limit, ok, because they are both defined by the sequence. I don't know how a limit can be said to have a limit...

-- Mindy

The reason I brought up binary notation (base 2) was that in that notation, the fact 1 = 1/2 + 1/4 + 1/8 + ... is written as 1 = 0.111..., by definition. So it'd be odd if you accepted a sum of 1 in one notation and not 1 one in another. I'm not sure why you bring up Archimedes' constant.

Anyway, as to your questions, there is no largest term of the sequence. The notation "0.999..." is just how everyone is (or at least how I am) writing "limit of the sequence (0.9, 0.99, 0.999, ...)" in shorthand. This limit has been shown to be the value 1, hence "0.999... = 1". If "0.999..." doesn't mean this, I don't know what else it would mean.

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Division is such a method. A quotient is obtained if and only if the process is completed. However, mathematicians discovered there were division problems that could not be completed. What to do? There are two possibilities. One is to say that division isn't defined in those situations, just as we say division isn't defined when the divisor is 0.

This is, in some sense, correct. However, this view suggests that we could, if we wanted, defined division by zero. And that is, in some sense, correct. But then we would not be working with arithmetic, i.e. the mathematical system meant to represent counting. Likewise when we define 5/3, we are no longer working with arithmetic. To extend our mathematical system in this way actually changes the domain in a deep way. Rational numbers do not count things except in the sense that the natural numbers "live inside of" the rationals, or to put it a little more formally, there's an injective homomorphism from the naturals to the rationals. So the development of the rational numbers from the naturals (or integers) is not one of patching some uncomfortable gaps but of constructing a genuinely new structure.

That is how negative signs came into use, and how the "...", etc. notation for non-terminating decimals came to be. (It is also how each new type of number came to be defined, Whole, Integers, Rational, etc.)

Particularly on account of the negatives, I challenge this claim. I don't believe negatives came about in order to allow us to extend the subtraction operation to a larger class of pairs of numbers, but would be much more willing to believe that it came about as an attempt to symbolize debt.

That means that any non-terminating decimal is explicitly not a quotient.

In what way? .333... = 1/3 in the real number system, by the argument I gave earlier. They are two names of the same entity in the structure, just as pi is another name for 3.14... which is another name for the ratio between the circumference and diameter of a circle. They are indistinguishable in their properties as mathematical entities.

Perhaps a large part of the problem here is the idea that the number .333... is somehow created by dividing one by three, approximated to the nearest tenth, then hundredth, then thousandth, and so on, and so the existence of the number is dependent upon how far out we actually go. However, the limit of a sequence is not constructed by us in that sense. For instance, take the sequence of numbers s_n = 1/n. Thus s_1 = 1, s_2 = 1/2, s_3 = 1/3, and so on. This sequence is not a finite sequence, and the sequence does not depend on how long we choose to state examples. Moreover, this sequence of elements approaches 0 in the limit. What I mean when I say that is nothing more than the claim: For every positive real number e, there exists a natural number N such that, for all natural numbers n that are greater than N, s_n < e. So give me any e, there is some rational number p/q which is between 0 and e where p and q have no common factor, then s_q = 1/q, which is less than or equal to p/q < e, thus I have proven my claim. That's all there is to claiming that some element is the limit of a sequence of elements. One doesn't have to talk about any infinite process at all, just infinite sets (Here, the sequence is an infinite set of rational numbers, and there is no process of generating the sequence. This is a defined set, regardless of whether anybody writes down or thinks of all the elements in it.). This notion of limit is essentially the same one that we use whenever we talk about any limits in analysis, so really the only objection that one might level against the use of real numbers is talk about infinite sets. But if you object to this, then you must object to the use of infinitely many natural numbers, and I presume nobody objects to that.

They do not qualify as numbers.

You should tell us what you mean by a "number" then, since in mathematics we simply take numbers to be elements in certain structures, usually ones that satisfy commutative unitary field axioms. Hence, complex numbers are numbers.

The argument that says the distance between .999... and 1 on the number line is infinitesimably small, so that they are not separate, is mistaking infinitesimals.

In the real number system, this is false. I think maybe in surreal numbers, this is true. But it is merely a consequence of the definition of real numbers, i.e. merely by the way that real numbers are constructed, that two real numbers are equal if and only if their distance is zero, and again by construction, distance between real numbers is zero iff the distance is arbitrarily small.

Since non-terminating decimals fail to be meaningful designations of quantity, they must be assigned a numerical value. .999... gets assigned 1. But that is an assignment of a quantity to a symbol, it is not a mathematical result. .999... is merely deemed equal to 1, it was not found to be equal to it.

This is just false. That .999... = 1 is not an assumption, it is a result which can be proved by properties of rational numbers and construction.

I am not sure how this relates to my post, but please note the in the case with .999... one never leaves the domain of rational numbers: all terms in the sequence are rational, at least some of the definition of the concept of limit are valid for rationals, the proof that the limit is 1 uses only rational numbers, and the result is also rational.

I speak about reals because the way to construct rationals from integers is just to take them as equivalence classes pairs of integers (where the right-hand part of the pair is non-zero). There's really no great leap from integers to rationals. There is a tremendous leap from the rationals to the reals, and that's where infinite decimal notation gets interesting. But yes, the numbers we've been discussing are contained in the rationals.

This is a very loose language. Nothing could be equal to an infinite sum, because there are no infinite anything, including sums. There are sequences of partial sums, which can have a limit. Something can be equal to that limit.

First, I've never understood this bizarre Objectivist insistence that nothing is infinite.

Second, see the above, when I speak of an infinite sum I'm not speaking of an infinite process--and keep in mind that numbers aren't things. When speaking about an infinite sum, one merely refers to the limit of a sequence of partial sums. No mathematician means anything else by this, so I can't see what your objection is.

The initial sense is one of a pragmatic approach.

It depends on what you mean by "pragmatic". Why do I cook food? Because it is the most pragmatic way for me to get nutrition. Being a practice is not in itself an objection. Mathematics, I claim, is just a practice. It does not, in and of itself, refer to anything.

As the concept of method is the 'how' we expand the numbers, and in continuous quantity, it permits the 'how' of refinement of resolution. This is done by setting aside for the moment the 'what' is being quantified or is being used as the unit.

We might use something like the concept of method to think about the numbers, but that's not what they are. An infinite set is still an infinite set regardless of whether anybody thinks of some or any of its elements.

The reason to suspect that the universe is finite in the context of quantity of electrons would be the law of identity. Treating arithmetic as oblivious, or as being agnostic seems to suggest a severing of the logical hierarchy between arithmetic and what perceptually gives rise to its necessity.

I never understood why Objectivist think that somehow identity is violated by an infinite quantity of something. It would just have, as part of its identity, an infinity quantity. Unless, of course, this law of identity is actually supposed to have some extra philosophical claim besides just that things behave in the way that they behave; that they are the things they are, and not anything else. Infinite things are infinite. Identity preserved.

And yes, arithmetic is not what historically gave rise to arithmetic. Arithmetic is a formal system with an infinity of elements, regardless of whether there are actually infinitely many objects in existence.

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I never understood why Objectivist think that somehow identity is violated by an infinite quantity of something. It would just have, as part of its identity, an infinity quantity. Unless, of course, this law of identity is actually supposed to have some extra philosophical claim besides just that things behave in the way that they behave; that they are the things they are, and not anything else. Infinite things are infinite. Identity preserved.

Infinity: "There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality." Leonard Peikoff, “The Philosophy of Objectivism” lecture series (1976), Lecture 3.

Things are what they are (Identity), and all existents are finite. There's no such thing as something with an "infinite quantity of something."

The law of identity doesn't mean that "things behave the way that they behave," and "infinite things are infinite" and therefore behave in an infinite manner consistent with their infinite nature.

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The reason I brought up binary notation (base 2) was that in that notation, the fact 1 = 1/2 + 1/4 + 1/8 + ... is written as 1 = 0.111..., by definition. So it'd be odd if you accepted a sum of 1 in one notation and not 1 one in another. I'm not sure why you bring up Archimedes' constant.

Anyway, as to your questions, there is no largest term of the sequence. The notation "0.999..." is just how everyone is (or at least how I am) writing "limit of the sequence (0.9, 0.99, 0.999, ...)" in shorthand. This limit has been shown to be the value 1, hence "0.999... = 1". If "0.999..." doesn't mean this, I don't know what else it would mean.

What else it would mean is a quotient, like 2/3 = .666... .

-- Mindy

p.s. Well said, Trebor (preceding post)

Edited by Mindy
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Infinity: "There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality." Leonard Peikoff, “The Philosophy of Objectivism” lecture series (1976), Lecture 3.

Things are what they are (Identity), and all existents are finite. There's no such thing as something with an "infinite quantity of something."

The law of identity doesn't mean that "things behave the way that they behave," and "infinite things are infinite" and therefore behave in an infinite manner consistent with their infinite nature.

Besides philosophical issues about mathematics we've already discussed, I do not assent to the premise that all existents are finite. I see no reason why that should be, and I see no reason why the law of identity in particular has anything to say about this particular matter since it merely tells us that one thing is itself and not another. If it is a part of a thing's identity to have an infinity of parts, in any sense, then it still retains identity and the law of identity says nothing substantial about this. So unless you can give a more explicit statement of the law of identity beyond A equals A, namely an explanation of how the law bears on the issue of infinite quantities, this road seems like a dead end. However, I do suggest that this discussion--if we're to pursue it further--be moved to a more appropriate topic. I'm sure there's one in the archives.

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Besides philosophical issues about mathematics we've already discussed, I do not assent to the premise that all existents are finite. I see no reason why that should be, and I see no reason why the law of identity in particular has anything to say about this particular matter since it merely tells us that one thing is itself and not another. If it is a part of a thing's identity to have an infinity of parts, in any sense, then it still retains identity and the law of identity says nothing substantial about this. So unless you can give a more explicit statement of the law of identity beyond A equals A, namely an explanation of how the law bears on the issue of infinite quantities, this road seems like a dead end. However, I do suggest that this discussion--if we're to pursue it further--be moved to a more appropriate topic. I'm sure there's one in the archives.

Let's see: Objectivism Online Forum.

Philosophy.

Metaphysics and Epistemology.

What does .9999999999… mean?

(subquestiona: What does infinity mean?)

What possible observations give rise to the concept of infinity in this case? (Hopefully this is taken as rhetorically as it was meant.)

What observations give rise to the necessity of either of these inter-related issues?

These items either reduce down to, that is create a logical heirarchy of the first level abstractions that give rise to them, or they would be invalid.

It is within the context of mathematics that give rise to both methods inquired about here. What gives them validity outside of the realm of mathematics? You wish to apply infinite to quantity. What gives rise to the concept of quantity? It is the relationship of a group of entities relative to one of its units. (Corvini, 2,3,4 and All That.) We can observe and grasp 2. We can observe and grasp 3, 4, and some of us 5, maybe 6. After that, we need a method to expand and deal with larger quantities.

We look out at the our world and see trees, and forests? We are able to extrapolate the square area of growable area on its surface. Would you suggest that an infinite amount of trees could be within a forest of that specific area?

We look at our galaxy. We observe stars. Via instruments and calculations we can assess its size. Would you suggest that our galaxy has an infinite amount of stars within it?

We look through microscopes and other instruments to conclude that our bodies are made up of cells. Would you conclude that our bodies have an infinite amount of cells within it?

Every observation that you are able to quantify and relate back to the existent that made it up - of them, have you discovered any entity without specific, finite, contextual limits?

Every system, and subsystem you are able to either observe directly, or relate the data back to the observable, have you discoverd any without specific, finite, contextual limits?

How many observations do you need to integrate before you permit yourself to induce the Law of Identity applies not only to the Universe as an entirety, but to each existent within it, within the framework of the context which gives rise to it, including the concept of infinity, and .9999999999…?

By chopping off the logical hierarchial chain part way down, and then trying to apply it within a context which it was not validated for, could account for some of the confusion that appears to be present here.

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Let's see: Objectivism Online Forum.

Philosophy.

Metaphysics and Epistemology.

What does .9999999999… mean?

(subquestiona: What does infinity mean?)

Well, if you think that .999..., thought of as possibly distinct from 1, is meant to name some quantity in the universe, I suppose this could be a germane place for the question, though I took that possibility to be more in the realm of excessive absurdity.

What possible observations give rise to the concept of infinity in this case? (Hopefully this is taken as rhetorically as it was meant.)

It won't be.

We observe that, for every quantity we can conceive, we can conceive of one greater. Thus, in order to handle all possible situations, we generalize to a mathematical structure which has an infinite domain.

What observations give rise to the necessity of either of these inter-related issues?

These items either reduce down to, that is create a logical heirarchy of the first level abstractions that give rise to them, or they would be invalid.

It is within the context of mathematics that give rise to both methods inquired about here. What gives them validity outside of the realm of mathematics?

These are clearly English words, but I can't make sense of how you've put them together.

You wish to apply infinite to quantity.

I do? I just state that there's nothing contradictory about the idea.

Would you suggest that an infinite amount of trees could be within a forest of that specific area?

If I thought that trees could be infinitely small then I might think it possible, but I've observed that they seem not to reach a size smaller than, say, their seeds (to be as generous as possible to the starting-point of a tree).

We look at our galaxy. We observe stars. Via instruments and calculations we can assess its size. Would you suggest that our galaxy has an infinite amount of stars within it?

If I had some good reason to, sure. I know of no reason to suppose it is impossible. I also don't know that there aren't arbitrarily small distances in space, or that each piece of matter is infinitely subdivisible into more basic units of matter (i.e. that matter is gunky).

How many observations do you need to integrate before you permit yourself to induce the Law of Identity applies not only to the Universe as an entirety, but to each existent within it, within the framework of the context which gives rise to it, including the concept of infinity, and .9999999999…?

I don't form my beliefs about possibility by some collection of observations, but by whether a hypothesis is self-contradictory. I form probability--not possibility--judgments on the basis of observations.

By chopping off the logical hierarchial chain part way down, and then trying to apply it within a context which it was not validated for, could account for some of the confusion that appears to be present here.

Again, I don't know what you're saying. The only logical hierarchy I've been discussing is within mathematics.

Edited by aleph_0
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Besides philosophical issues about mathematics we've already discussed, I do not assent to the premise that all existents are finite. I see no reason why that should be, and I see no reason why the law of identity in particular has anything to say about this particular matter since it merely tells us that one thing is itself and not another. If it is a part of a thing's identity to have an infinity of parts, in any sense, then it still retains identity and the law of identity says nothing substantial about this. So unless you can give a more explicit statement of the law of identity beyond A equals A, namely an explanation of how the law bears on the issue of infinite quantities, this road seems like a dead end. However, I do suggest that this discussion--if we're to pursue it further--be moved to a more appropriate topic. I'm sure there's one in the archives.

The Law of Identity is not some man-made decree or a lame attempt to impose some arbitrary, short-sighted limitation on reality. It is a grasp of a fundamental, self-evident fact of reality.

"Axiomatic Concepts"

"Identity."

"Infinity"

Edited by Trebor
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To add some perspective to the problem of "0.999..." :

Taking a fraction, one obtains its decimal representation by performing the long division, and one finds out, most of the time, that it never ends. However, in the case it never ends, there is a sequence of one or more digits which repeat themselves indefinitely. For example:

1/3 gives 0.3333 ...

7/9 gives 0.7777 ...

5/7 gives 0.714285714285 ... (repeating 714285)

124/555 gives 0.22342342 ... (repeating 342 after the initial 22)

(I've used the calculator, of course :-)

Interestingly, the property of a fraction to result in an unending representation is not an intrinsic property of that fraction: in a representation other than decimal (base 10) we could get a terminating representation.

For example, in the base 3 numbering system, 1/3=0.1 and 7/9=0.21. In base 7, 5/7=0.5. I don't know in which base 124/555 gives a terminating dot representation, but I recommend the Wikipedia article Repeating Decimal for some very interesting facts. Among other such facts, this article notes that even the terminating representations, like 0.57, can be also represented as two nonterminating ones:

0.57= 0.57000...=0.56999...

(the terminating ones are terminated because, by convention, one suppresses the trailing zeroes).

Anyway, all of the above shows that we shouldn't read too much into the fact that decimal dot representations of some fractions are unending.

Sasha

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The Law of Identity is not some man-made decree or a lame attempt to impose some arbitrary, short-sighted limitation on reality. It is a grasp of a fundamental, self-evident fact of reality.

"Axiomatic Concepts"

"Identity."

"Infinity"

I've read all of Rand's books and OPAR, so unless these links contain some arguments which are not contained in the literature, I'm afraid I won't find them informative.

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I've read all of Rand's books and OPAR, so unless these links contain some arguments which are not contained in the literature, I'm afraid I won't find them informative.

Glad to hear that you reject the idea that there can be either infinite existents or characteristics!

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