Because it’s not true. It’s not equal. 0.333… isn’t even a number.
Sounds like a major problem.
Well, either 0.999… =1 or it doesn’t (as Iambig might say) and that has some effect on math results. You can’t just wave it off. You can’t wave off a fundamental problem with division (1/3 =/= 0.333…). Where is the rigor then?
Coherent???
You just whacked a bunch of coherence in math, if your analysis is correct. You just broke a bunch of math properties.
Being pedantic, you are correct. The exact truth is that 1/3 => 0.333… {leads to, not equates to}
But no one has serious practical reason to care. And an argument from pragmatism is fallacious. Truth doesn’t depend on how practical the conclusion might be for society (except to the priesthood).
The only incoherence being corrected in common math leads to the inclusion of the proven hyperreal system. And the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities. Such can be merely stated in a foot note. Nothing very serious changes anywhere.
The provable incoherences in Relativity Ontology and especially Quantum Physics are far more serious.
Of course they are about quantities - infinities and infinitesimals become usable quantities within the system. That’s the whole point, that infinity and infinitesimal are no longer hand-wavy concepts.
Degree of infinity? What??
That’s what Wtf says that the transfer principle produces. You haven’t disproved his assertion.
“degree of infinity” , “degree of infinitesimal” - What is that??
By “feel” I was referring to intuition, that sense that something is a particular way because similar things are that particular way. The infinite series is merely the mathematical equivalent to the word “infinity” itself: the word allows us to name that which has not and cannot be achieved in the material universe but surely exists (yes, I know this leads to another conversation entirely!), while the series allows us to measure that which is endless. Mathematics is the art of measurement, after all.
The infinite series is not smaller because “there is ALWAYS a smaller amount to be added” because THAT is not what the infinite series IS. That kind of thinking implies a truncation: one has looked NOT at the series as a whole but only up to a point…and found it wanting. Just as we say “infinity” to name something endless, the series must be seen as a single name for the process.
Just because the two propositions are not differentiable, does not mean they are not whole, meaning by logic they fall into each other. Wheather that kind of union consist of some interpretation of what ‘whole’ means, is a matter of definition.
There is no such a thing as “infinite set”. That’s just poetry.
What there are are algorithms – not sets – that have no exit point. This means algorithms that do not tell you when to stop following their instructions.
The algorithm that tells you how to list real numbers is one such algorithm.
Yes, I understood what you meant. And my reply still stands.
But do you not realize that “infinity” does not exist period, not merely in the physical, but even the concept is an error (produced by those who rely too much on the intuition). That which is infinite has no end. The concept of “infinity” is the concept of “The greatest point of that which has no greatest point” or “the end of endlessness” - an oxymoron.
The “state of being infinite” is called “infinite”, not called “infinity”. But they had to make up something because people keep using the word as if the noun actually made sense. The word “infinite” is the adjective, not “infinity”.
And then in math, obviously there is no number greater than any number that can be applied to an infinite series. That is the oxymoron in the common usage.
More or less true. Mathematics is logic as applied to quantities. Science is more the art of measurement.
This is the part that is interesting and again, seemingly self-contradictory.
“The infinite series is not smaller because “there is ALWAYS a smaller amount to be added” because THAT is not what the infinite series IS.”
It’s hard to be certain, but I think you’re stating that an infinite series is not a series that always has more to add. That would be an error in definition on your part. The ellipsis, “…” specifically means “infinitely/endlessly extended”, having no end or destination. The word “infinity” implies an final destination.
“That kind of thinking implies a truncation:”
That is the contrary part. Thinking that something is endless, is not implying a truncation at all, quite the reverse. Thinking that infinity exists, even as a valid concept, is implying a truncation. It implies that there is a point “at infinity”, yet there is no “at infinity”. Those who think that 1 = 0.999… are thinking in terms of “once the series reaches infinity, it will be equivalent to 1.0”. But of course, there is no “infinity” to be reached. They “truncate” the endless process.
“one has looked NOT at the series as a whole but only up to a point…and found it wanting.”
One has looked at what the series states and found that there is no end and thus “found it wanting”. The more simple minded imagines an end “at infinity” and thus sees it as a completed process. It is a common error of imagination.
“Just as we say “infinity” to name something endless,”
That part would be largely true, but more like “we say ‘infinity’ to name the destination of something endless”. And that of course, is logically senseless, but intuitively implies the greatest possible even though there is no “greatest possible”.
“the series must be seen as a single name for the process.”
An ENDLESS process, yes.
The point is that an infinite set may be closed, as between .9 and 1, where the terms .9+.09+.009 are the members of the set. That this series can be interpreted as an infinitely divisible set, (by definition), has no linguistic barrier, that declares that such a proposal is logically unsound. Between logic and language there is this unassailable affinity.
On the other hand, the same set is bounded , by 0, and infinity, an infinity which again is a conceptual bind, vis: where the bind is the antimony, or the contradiction within it’s own meaning structure.
For instance, the infinite is contradicted by the finite, there can not be a definitionally infinite, without the positing of the finite structures abiding within the structure of language it’self.
Now comes the usual critique as to the reality of the logic of language, as it corresponds to reality. Does it?
Does language create reality, or conversely does reality create language? Or, does it really matter, or even make sense to pose this question in such terms?
Linguists would say that it is language and logic which need to correspond to reality, and I don’t think they say anything else, or whether it matters at all which takes precedence.
Why? Because of the spational -temporal relativity, but that is a conclusion, and sorrily, I have to argue backwards, and if you do, you have to come to view spational relationships as formative. I really do not want to get into this , though, because of a belief in the intuitive basis of math, is as suspect as it is esoteric, almost borderline occult.
Every algorithm is defined by some set of rules which means that the range of possible outcomes is limited (as you say, closed.) Otherwise, it wouldn’t be much of an algorithm, right? It would be an absence of rules, and thus an absence of algorithm, which can be poetically expressed as an all-permitting pseudo-algorithm that says “do whatever you want”.
If an algorithm, which is basically a set of restrictions, is such that it has a definite number of permitted values, then it can be represented using a set. However, if an algorithm is such that it does not have a definite number of permitted values, then it cannot be represented using a set, only poetically using the idea of “infinite set” which is just a reflection of our desire for something that isn’t there.
Nonetheless, even though the number of permitted values is indefinite, we can still, somehow, determine which among such algorithms has the greatest number of permitted values. The question is how is that possible if we do not have the corresponding sets?
The answer is because we are not comparing sets of permitted values (which are non-existent) but sets of restrictions that define these algorithms (which are existent.)
The algorithm that has the least number of restrictions – the least restrictive algorithm – is said to have the greatest number of permitted values.
If you have a rule that says “do anything except for lying” and a rule that says “do anything except for lying and stealing” the former rule is clearly less restrictive and therefore more permissive which means it has a greater number of permitted values (in this case, values would be behaviors.)
0.999… is strictly speaking an algorithm without an exit point that generates a number. You need to force quit it to get the final number (the algorithm does not produce one) or the algorithm will proceed generating the number by making it closer to 1 but never 1.
How can an algorithm be a number?
Especially one that does not have an exit point?
Unless you plug it into some function (such as limitOf) that returns a number. But then, it would be f(algorithm)=number and not algorithm=number.
Sorry I only have my Samsung phone to write with. But generally yes you can talk or think in terms of an infinite set ,non poetically, a set of numbers is a set whether it is finite or not.
Now we could refer to James in saying that infinity as a concept or a set of numbers does not exist, but the only boundary which separates a finite set of numbers from an infinite one, is the idea behind putting a limit on the number of a decimal, after which there is no functionally derived significance.
But the problem with that is, that such a value Is incalculable.
Therefore, the calculus of differentials can not solve that problem, inasmuch as Outside the boundary of the problem.
It is not a matter of the existence of infinity, because, the tools by which it can be shown to ‘exist’ have no such capacity, except to the degree there is a function to which there is a derivative.
Therefore, it becomes a matter of an intuitive
process, the idea of which goes back to Meno,
another paradoxical Platonic/Socratic character.
The first step would be to establish whether the concept of infinite makes any sense. Without this step, no discussion can take place.
My contention is that there is NO such a thing. Noone ever observed anything that is without an end. The act of observation requires that what is observed has an end.
