Jerkey, your use of Hume’s guillotine appears to have killed this thread.
Not so much.
Carleas only stops arguing when he has been embarrassed into a corner and not released. Carleas stopped arguing. Phyllo always steps in when he can find something to point out as wrong (besides merely the contended conclusion). Phyllo stopped stepping in. Wtf is bright enough to see when he no longer has a valid argument (even still doesn’t want to give in). Wtf couldn’t find any remaining argument to claim as valid. Uccisore, knowing almost nothing of this topic, attempts to help out the contemporary social authorities against what he perceives as “the horde”, but gets frustrated and goes berserk. If given the chance to calm down, he then leaves.
The upshot is that they could not find anything invalid with the proofs that 1.0 is NOT equal to 0.999… And on this forum, no one yields so as to cause real resolution (where would be the hell in that). So they just stop posting.
Who are these “contemporary social authorities” you speak of?
One can say that for a specific purpose, in a narrow, prescribed way .999 is “virtually” equal to one. If .999 was actually equal to 1 then it would not be .999 but 1; simple as that.
Where a philosopher might agree with the mathematician is in denying the actual reality or experience of a 1. Formally, sure, but in practice time means change and change is a challenge to a permanent identity
[quote=“One Liner”]
Jerkey, your use of Hume’s guillotine appears to have killed this thread.[/
Rather then killing the forum, it may add another essential dimension.
I think JSS has been clever in his approach in that he never gives away too much information and relies on others to do their own research.
I’m truly amazed and disappointed in all of you that more people here believe 0.999… doesn’t equal 1 than do. Despite the literal mathematical proof.
It must be exhausting being so edgy all the time.
Which proof?
How can anybody who is trying to use logic ever “prove” that 0.999… = 1?
Where as Godel and other Neo-Kantians certainly make a good case for an argument for a structural set theory based on logical basis of intuitive mathematical prof that there is never a formal equality.
The hyperreals could never help. We were talking about the hyperreals (which are the real numbers in nonstandard analysis) back on page 14 and thereabouts. viewtopic.php?f=4&t=190558&start=325 I recommend that discussion to anyone who’s heard of the hyperreals and isn’t sure whether they pertain to the question of whether .999… = 1.
The bottom line is that the statement that .999… = 1 has the same truth value in the standard real numbers as it does in the hyperreals. That’s because the hyperreals are an alternate model of the same collection of axioms as the reals are. There’s a theorem called the transfer principle that says that any first-order statement is true in the reals if any only if it’s true in the hyperreals.
If we take the string “999… = 1” and drill it down to a statement in pure logic and set theory, we’d find that it’s a first-order statement. So if it’s true in the reals, it’s true in the hyperreals; and if it’s false in the reals, it’s false in the hyperreals.
We conclude that bringing in the hyperreals and nonstandard analysis is a tremendous distraction. People trying to understand why “.999… = 1” is a valid statement in math, would be far better off learning about what the real numbers and limits are; and not wandering off into tempting but unproductive digressions like the hyperreals.
His elementary school teachers in this case, it seems. FJ is the same way - "whatever I was taught in youth is holy doctrine".
The confusion regarding this issue seems to be related directly to an ontology conflation that Carleas pointed out as “Zeno’s Paradox”. To Carleas’ credit, Zeno can be viewed from 2 perspectives; one concerning motion, as Aristotle understood him to mean and the other concerning the ontological definition of distance. Once you get those straight, ALL of the issues with calculus, infinities, and infinitesimals go away. For ages, the confusion has merely been a trick of the mind.
I’ll post a thread on Zeno shortly to hopefully clear it up.
I agree that the discussion of hyperreals is not necessary, although the issue is exactly on the border between the standard reals and the hyperreals. In the long run it is merely an ontology issue.
comment withdrawn
It may not be necessary but it points us in the direction of an answer.
Obviously people do not know what the real numbers and limits are and so how do we move forward.
All you are saying is that 0.999. . . is the limit of the sequence .9, .99, .999, etc and hence is always less than one (it’s not rocket science).
Let’s take a whole. Naturally, we are inclined to represent this whole with a symbol of ONE.
Now let’s divide this whole into 10 pieces. Naturally, we are inclined to represent the result of this division with a symbol of 1/10, but this is not a good idea because it leads to confusion. Mathematical division works differently. In mathematics, division refers to a division of multiplicity, not to a division of singularity. Mathematics is about taking a multiplicity and then distributing it equally across a number of divisions. The result of mathematical division is a number of pieces contained within each division. Number one isn’t really a multiplicity, so if we are to represent it using mathematical division we will have to convert it to some sort of multiplicity. Because we are dividing by 10, let this multiplicity be 10. Thus, let our whole divided into 10 pieces be represented by mathematical division as 10/10.
Now let’s isolate a piece. How would we represent this piece? Mathematically, we would represent it as 1/10. We can also represent it as 0.1.
When we isolate a piece what we get is a situation of one isolated piece and nine pieces that are not isolated. This can be represented as 1/10 + 9/10 or 0.1 + 0.9.
Now let’s take this isolated piece and divide it into 10 pieces. What do we get? We get ten new pieces. Each one of these pieces is now the original isolated piece divided by 10. This means 1/10 divided by 10 which is 1/100 or 0.01.
The situation is now as such: 10 x 0.01 + 0.9.
Now let’s isolate a piece from this set of ten new pieces. What do we get? We get 0.01 + 9 x 0.01 + 0.9, or 0.01 + 0.09 + 0.9, or quite simply 0.01 + 0.99.
Now if we repeated the cycle one more time what we would get is 0.001 + 0.999.
And if we decided to repeat this infinitely, how would we represent it?
We would represent it as . . .
0.999…
Notice what we did here: we started with 1, then went over a number of re-expressions of the same symbol, finally ending up with 0.999…
0.999… is equal to 1.
If you want to freeze 0.999… into a finite expression, then you will have to add the remainder in order to make it work. That is a given.
So what is the problem with people who deny that 0.999… equals 1?
I suppose then that no number is equal to itself because of their infinitely trailing zeros that never allow them to resolve to a value . . .
You were doing fine up to that point.
Notice that before your conclusion, you had;
1 = 0.1 + 0.9
1 = 0.01 + 0.99
1 = 0.001 + 0.999
.
.
but then you switched so as to leave off the initial tidbit:
1 = ______ + 0.999…
Using that kind of representation, it would really have to be:
1 = 0.000…1 + 0.999…
or
wherein R = “ever Remaining infinitesimal”
[1.000…:0R] = [0.000…:1R] + [0.999…:9R]
Or as I stated it long ago:
The “problem” is that they/we don’t like leaving off that remainder that you just mentioned was necessary.
A zero represents nothing. So in those cases, they are leaving off nothing.