Only through an intuition of math/reason, can a reductive differentiation of two types of logic manifest a connection.
Difference through similarity-familiarity, and identity through reason if, not exclusive, then at some point may share a functional derivative. The difference at both absolute limits, are somehow limited without violating the definition of the limitless. They are at once limited and limitless at this point, and in spite of contradiction of nomenclature such contradiction resolves. Perhaps this is because the way the contradixtion effects and changes, as do quantum effects seem to change by the effected quantum particles.
Because of Carleas objection of not introducing non mathematical considerations, i withdraw my response to James quiery into logical contradixtion, but let stand as a valid conjecture holding that a purely mathematical demonstration will lead into that very contradiction.
Here, we have to be careful. InfA is not a number on the standard real line. It’s true that the set of decimal places of .333… is of the same cardinality as the set of integers. But calling that cardinality a “quantity” or a “number” are being a little too loose with language, especially since it looks like you mean to build a rigorous syllogism. I applaud that effort, but if that’s the goal, we should be wary of accidentally introducing ambiguities.
Does that answer the question, or if not, is there a way to reword the question to address my concerns? Or perhaps my concerns are misplaced?
We have previously agreed that infinites can be of different sizes. One particular size can be ("may validly be ") set as a standard with which to compare others. Obviously if there are different sizes, there are differences between those sizes to be potentially measured.
I am giving one particular infinite size a name, “infA”, to be the size of the set of all integers. InfA does not have to be called a “number” or a “real number” or a “quantity” or anything other than a standard size (aka “infA elements”) of the infinite set of integers.
So we have agreed on the 2nd question of the set of digits to the right of the decimal being of size infA. Or that there are “infA elements in the set”.
Therein lies your fault. You treat something that is not a number as a number. Which makes you think that there is a zero at a particular position on the end. In fact, there is no end.
Yeah, I’m with Phyllo here too. Your reluctance to talk about it as a cardinality rather than a size suggests that you mean something different. Is that right? My concern is that when you call it a ‘size’, there will be a temptation to make an equation like
InfA + 1 > InfA
which is not the case if InfA is the cardinality of the integers (and 1 is the cardinality of a set with 1 element).
And again, I mean to be pedantic here, because I expect the syllogism you are building to be sensitive to the details. Cardinality is a specific mathematical concept; ‘size’ is ambiguous. The set of integers has the same cardinality as the set of decimal places of .333… If you mean something different by size, you need to be explicit.
You cite to non-technical sources, so we should expect those definitions to be a bit blunt for many purposes, and they are for this discussion: Infinity isn’t a number, so the cardinality of an infinite set can’t simply be the “number of elements” (though it is for finite sets). And infinite sets and their cardinalities have weird properties that make treating them as numbers problematic; for example,
InfA + 1 = InfA (where InfA is the cardinality of the integers, and 1 is the cardinality of a set with one element).
By contrast, if 1 and 3 are the cardinalities of sets with 1 and 3 elements, then
1 + 3 = 4
where the + symbol in relation to cardinalities is the disjoint union operation (if it is a regular union, then 1 + 3 <= 4)
If all of this is accepted, and you are just using “size” to mean “cardinality”, then proceed. Otherwise, could you clarify how they are different?
Certainly not in the sense that it is being used in general math equations.
An infinite series, or sum, or integral does not specify some particular infinity as opposed to another infinity.
The reason seems intuitively obvious - the very concept of ‘infinity’ does away with all size concerns. What applies for the “lowest” infinity also applies to all “higher” infinities. IOW, the results for all infinities are exactly the same.
I think that intuition is right for limits, series, sums, etc., but there is a sense in which there are ‘more’ real numbers than natural numbers, while there are the same number of natural numbers and integers. You can map uniquely from the natural numbers to the integers and vice versa (e.g. by mapping 1 to 0, 2 to 1, 3 to -1, 4 to 2, 5 to -2, etc.). But the real numbers are uncountably infinite: You can’t create a mapping, because there’s no next number for any real number; any two real numbers have infinitely many real numbers between them.
I don’t know why you guys are going off on this tangent.
Certainly, there is no reason for James to do so. It weakens his “you can never get to the end” argument. A number system with cardinality infA will have gaps between numbers as compared to a system with greater cardinality infB. There will obviously be a ‘smallest number’ in the system. That’s something he denied was true for Reals.
I wish to clarify whether or not you mean cardinality when you say size, since as your own source points out, “[t]he cardinality of a set is also called its size, when no confusion with other notions of size is possible” (emphasis added). Just be explicit and we can move on.
You made the point yourself:
The existence of different infinities makes this a possibility. And if it is a different string, with a smaller infinity of decimal places, then James could refute proofs that rely on multiplying .333… by 10.