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Hippasos
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Does this reasoning ever reach infinity?
0<1<2<3<4<5...
What does it mean?
Thanks!
0<1<2<3<4<5...
What does it mean?
Thanks!
Last edited:
I think it means that n<n+1 as n goes towards infinity.Hippasos said:Does this reasoning ever reach infinity?
0<1<2<3<4<5...
What does it mean?
Thanks!
Hippasos said:So the chain
0<1<2<3...
1. Never ends while reaching infinity?
2. Never ends and does not ever reach infinity?
Hippasos said:" DaveC426913;2765460... Infinity is a concept, not a number ..."
That is interesting - so mathematics is not strictly about numbers but also concepts.
1. Mathematics without infinities i.e. concepts - not possible?
2. Mathematics does/does not require concepts to exist?
3. How do we know if a concept is a mathematical one?
Hippasos said:Does this reasoning ever reach infinity?
0<1<2<3<4<5...
What does it mean?
Thanks!
Hippasos said:Does this reasoning ever reach infinity?
0<1<2<3<4<5...
What does it mean?
Thanks!
Specifically:Hippasos said:Does this reasoning ever reach infinity?
0<1<2<3<4<5...
What does it mean?
Thanks!
Hurkyl said:e.g. what reasoning is the opening poster referring to? What does he mean by reach? And what exactly is the ellipsis covering up?
Specifically:
x < y < z is a mild form of 'abuse of notation' as it's called.
Well, I'm afraid it's not that simple because of a few reasons.Jamma said:So let's agree that x < y < z to mean x < y /\ y < z, x < y < z < a to mean x < y /\ y < z /\ z < a ...
Just to be clear -- this comment is not aimed at finite strings of chained inequalities, but instead to the infinite string, right?I'm not saying it's not possible to make it rigorous, or even to just work with it under the assumption of understanding, I'm just pointing out that technically it's as unmathematical as saying such things as 'the infinitieth digit 0.999...'
Since I have no idea what you're talking about, probably neither.Hurkyl said:Just to be clear -- this comment is not aimed at finite strings of chained inequalities, but instead to the infinite string, right?
It makes as much sense as things like N := {1,2,3 ...}. And it happens to be true.Jarle said:0 < 1 /\ 1 < 2 /\ 2 < 3 ... is a reasonable interpretation of 0<1<2<3..., but what does this mean? It is supposed to represent an infinite string of symbols, but that doesn't make much sense. What you want to say can be described by using quantifiers as such: [tex]\forall n \in \mathbb{Z}^{+} (n-1 < n)[/tex].
ZQrn said:It makes as much sense as things like N := {1,2,3 ...}. And it happens to be true.
The expression 0 < 1 /\ 1 < 2 /\ 2 < 3 ... is simply true, why, because true is a value, and the limit of that expression converges on that value.
ZQrn said:Well, I'm afraid it's not that simple because of a few reasons.
1: Context free grammar, if we define it like this then mathematical notation is no longer generated by a context free grammar and violates the basic rule that x # y $ z is either (x # y) $ z or x # (y $ z). Which it is can be inferred from the associativity and the precedences of operators.
2: Formalism, where 'meaning' is defined as simple manipulations of symbols, it gets a lot more complex to define the inference if it's not context free.
3: ambiguity, the point is the 'false' is strictly smaller than 'true'. Indeed, we call the mathematical relationships of disjunction and conjunction monotonous because in disjunction the result is always equal or greater than its operants, and in conjunction it's always equal or less than its operants. So a < b < c < d would technically mean (a < b) < (c < d). Essentially implying here in a system of binary logic that the the former subpart is false, and the latter is true. This may be useless in binary logic, but in modal logic or fuzzy logic this has more implications.
I'm not saying it's not possible to make it rigorous, or even to just work with it under the assumption of understanding, I'm just pointing out that technically it's as unmathematical as saying such things as 'the infinitieth digit 0.999...'
See Hurkyl's post, context free is a technical term. Mathematics for the most part is context free and mathematical notation is clearly intended as such.Jamma said:Did I not make it rigourous (although the end of my post was meant to be more of a quirky joke, I guess I appreciate that it wasn't funny now )?
I guess you're first point is valid, but maths isn't context free, you later used the word "lim" but the "i" in the middle no longer means the imaginary unit "i", neither are you implying that "lim" means "l*i*m". Maths isn't context free when there is no danger in it being so, and rightly so, imagine how long texts would have to be if it was always context free, and how many new symbols we would need. Being this picky isn't as logical as you think, if anything, it is stupid.
And saying that, couldn't you logically say that the "< < < < <" is a giant symbol in which the elements reside? I see no problem in this, the topological propoerty of connectedness isn't a requirement of a mathematical symbol, as the letter i proves.
You could argue that then we cannot tell if each individual "<" should be interpretted as part of the "< < < < <" or on its own, but then you can't tell when the "." of an "i" should be seen as a "." above an iota", but I suppose it's clear from the context isn't it?
I'm not saying that it makes sense, I'm saying that it's possible, nothing stops us from doing so.Jarle said:In what context can you make sense of limiting truth values?
EDIT: bad example, I withdraw it
To extend predicates over an infinite domain we will need a logical machinery such as set theory in order to make sense of them. In which case we use quantifiers such as [tex]\forall[/tex], and not an infinite string of symbols. N = {1,2,3...} makes sense because we know exactly what it is when we write it, not because it is a representation of an infinitely long sequence of consecutive integers.
ZQrn said:I actually realized it's already there, there is a total order on truth values, it's called logical implication, when we say [itex]P \rightarrow Q[/itex], we say precisely [itex]P \leq Q[/itex].
Really? I had no idea.yossell said:Just another nitpicky note, but logical implication is not the same as [itex]P \rightarrow Q[/itex], which is usually called material implication. Logical implication is not a total order: P does not logically imply Q nor does Q logically imply P, though it is true that one of [itex]P \rightarrow Q[/itex] or [itex]Q \rightarrow P[/itex] is always true.
ZQrn said:Then what is logical implication if different from this 'material implication'?
Ahh, so logical implication ranges over formulae, id est, expressions which resolve to truth values, while material implication ranges over variables which hold truth values?yossell said:At the propositional level, A logically implies B if and only if `A materially implies B' is *valid*.
So: (P & P -> Q) logically implies Q.
In general, given two propositional formulas A and B, A logically implies B iff, when you write out the full truth tables for A and B, there is no row of the truth table in which A is true and B is false.
In order to know whether A materially implies B, you have to know something about the actual truth values of A and B - you can't answer the question if you don't know either of the actual truth values. However, you don't have to consider any of the possible truth values.
In order to know whether A logically implies B, you don't have to know the actual truth values of A and B at all - however, you do have to look at the various possible truth values that A and B could have - by doing a truth-table - to answer the question.
ZQrn said:Ahh, so logical implication ranges over formulae, id est, expressions which resolve to truth values, while material implication ranges over variables which hold truth values?
Yes, but in our formal system of 'the real world', surely, everything implies that 'grass is green', because that sentence / formula is invariably true. We could say it's an axiom.yossell said:No, I didn't mean to imply that. Suppose we work with concrete sentences: Snow is white, Grass is green. Then 'Snow is white -> Grass is green' is true - for both consequent and antecedant are true, but `Snow is white' doesn't logically imply `Grass is green' for, since both sentences are atomic, there is a row in the truth table which makes the premise true and the conclusion false. However, `Snow is white -> Grass is green & Snow is white' does logically imply `Grass is green.' and even somebody who didn't know what snow was white and grass was green can know this - for by looking at all the rows in the 4-place truth table for these sentences, we can see that there is no row where the premise is true and the conclusion false.
I understand that they are different, I'm just pointing out that their definition seem to imply that they are incarnations of the same umbrella where the former deals with constants and the latter with bound variables.yossell said:No.
I've outlined two distinct concepts. You should be able to see, given my particular explanation, why it is not true that everything logically implies `grass is green.' Now, I don't mind if you say, that's not what you meant by 'logical' implication. But I can't see a way to make progress if you don't understand that there are two concepts at play.
I understand that some people make that distinction 'naively', yes, and what it would conceptually mean. The point is that 'tautologies' strictly speaking do not exist, there is no sentence which is 'universally true'.Do you understand the difference between (a) 'A -> B' is true, and (b) 'A -> B' is a truth-functional tautology?
Yes, this reasoning is valid. It follows the basic principles of mathematical reasoning and is a commonly used method of representing an infinite sequence.
Yes, this reasoning can continue indefinitely. As long as we keep adding 1 to the previous number, the sequence will continue to increase without limit.
No, there is no limit to how high the numbers can go in this sequence. As we continue to add 1 to the previous number, the numbers will continue to increase without bound.
Yes, this reasoning can be applied to any set of numbers as long as they follow the same pattern of increasing by 1. For example, 0.5<1.5<2.5<3.5<4.5<5.5 or -1<-0.5<0<0.5<1<1.5.
This reasoning is commonly used in mathematics and physics to represent infinite sequences, such as in the concept of limits and in the study of infinite series. It is also used in computer programming and data analysis to represent and manipulate large sets of numbers.