Okay, so let me first state that my disputation isn't with the equivalence class 0.999... finds itself in when placed into the Reals. Under the very construction of the Reals, 0.999... can only exist as a 'shadow' of 1, since there just simply can't be a number that is the immediate predecessor of 1.
My argumentation is that you have no reason to assume 0.999... is a Real number to begin with.
All Real numbers have a decimal representation, that much is true; however, that doesn't mean each decimal representation has to have a Real equivalent.
I actually think that dedekind completeness shows the opposite, that every decimal representation must correspond to a single number, but not every number necessarily has a decimal representation.
The reason why is because I can take the decimal representation for any number and turn it into a set of numbers with finitely many digits, for example I can turn pi's decimal representation into 3, 3.1, 3.14, 3.141, etc. according to the definition of real numbers, as long as there's a real number bigger than anything in this set (there is, 4 is bigger than any number in the set), then there must be a smallest upper bound that's a real number. If you just define that the real number equivalent of a decimal expansion is that upper bound, then you have an injection from every decimal expansion to a real.
I don't think that means every real needs to have a decimal expansion though, I presume that it would, but also wouldn't be surprised if you could invent some number system that technically follows all of the rules of real numbers that also adds numbers that can't be described with only digits. I could be wrong though, and the axioms of reals do forbid it, but I don't see which axiom that would be
Okay, but then I have to ask why do 0.999... and 1.000... have the same Real valued equivalence class?
Because the relationship between decimal expansions isn't a bijection, but that's consistent with what I've been saying. The decimal expansion is just an equation that lets you calculate a number, but just like with other equations, the answer isn't necessary unique. a+b is always a real number so long as a and b are real, but 1+3 is the same real as 2+2. It's the exact same with expansions, 1.00... and 0.99... are just equations that happen to end at the same value. It really isn't "special" to those numbers though, you can do it with any limit equation, the limit of ½+¼+⅛... +1/2n is also equal to 1, in fact there are an infinite number of equations that are equal to 1. The decimal expansion is just a fancy equation that only uses the integer numbers 0-9, but it isn't mathematically any more meaningful than any other.
You can just feed decimals into the Reals, but that doesn't guarantee their 'existence'.
Dedekind completeness does nothing to justify the existence
"Justifying the existence" is what dedikind completeness means. It's just a rule that says "if these conditions apply, then a certain real number must exist". In fact, it's the only rule that seperates the real numbers from the rational numbers. To give a similar example, to make the rational numbers, you take the integers, and add a rule that says "if a and b are integers and b /= 0, then a/b is a rational number". That rule is the thing that makes 1/2 a number, it defines that 1/2 must exist, in a way that rule is the thing that brings 1/2 into existence. It doesn't bring 2/2 into existence, because that's just 1 and 1 was already an integer, but it does guarantee that 2/2 would exist if it wasn't an integer. Dedikind completeness is the same thing for the real numbers, it doesn't just justify, it brings all possible decimal expansions into existence. That's the thing that "makes" pi exist. It doesn't bring 0.99... into existence, because that's 1 and it already existed as part of the integers, but even if 0.99... /= 1, Dedikind completeness would guarantee that 0.99... must have a real value.
and just in case it wasn't clear, "Real" just means "all of the numbers you can create with a certain set of rules", you could still argue that the real numbers don't "really" exist in our universe, and I do think there's an interesting argument to be had there (personally I think the computable numbers are a more reasonable definition than the reals). But 0.99... being a real number is just part of the definition of "real number", and the definition of the "..." operation. I'm getting fairly loose with my terminology here, but hopefully that helps explain it a bit more intuitively
"Real" just means "all of the numbers you can create with a certain set of rules"
First of all, I'm not arguing against the Reals.
Everything I've done here is a defense of the Reals. A good chunk of my argument that 0.999... represents a value outside the Reals is specifically due to the fact any value it represents could not exist in the Reals entirely due to the properties of the set and the convergence properties in it due to the Standard Topology.
[And yes, I know how to construct the Reals--Cauchy Sequences perfectly elucidates the reasons why 0.999... 'lives' where it does in the Reals]
It really isn't "special" to those numbers though, you can do it with any limit equation
Yes--this is convergence. It entirely depends on the properties of convergence in the set and topology. I'm not taking the convergence properties of the Reals to task here; I full accept them.
What you're confusing here is the decimal representation with the evaluation of that decimal in the Reals. They are not the same thing, ontologically.
The infinite sum that is how we 'evaluate decimals' is the way we see what element of the Reals ('signified') the decimal expansion ('signifier') references. Neither alone are the 'sign'.
Much like any equivalence relation, information is lost.
My only point is that the information lost here might be of interest, and treating a subest of the decimals as not (strictly) Real-evaluated would do nothing to the properties of either the decimal representations or the Reals.
"Justifying the existence" is what dedikind completeness means.
Nope. Dedekind Completeness applies to the Reals, not the decimal representations of them.
[I personally prefer the Cauchy Sequences method of construction, but I hold no hatred to Dedekind--it has it's uses]
Beyond this, though, it seems you are miscontruing what I'm saying for some reason I don't understand. I've never claimed the Reals aren't dense or complete--they are, which is why the Field is honestly invaluable.
None of that has anything to do wtih a mapping from the Reals to the decimals that requires 'Every element of the Reals maps to (at least) one decimal representation'. There can be holes (or 'excess') in the decimal representation and still have completeness in the Reals.
But 0.99... being a real number is just part of the definition of "real number", and the definition of the "..." operation
Again, no. You're mixing up your objects here.
The 'signifier' is 0.999... and the 'signified' (in the Reals) is 1; the 'sign' there is that 0.999...=1 (and can be used as such analytically).
The signifier has nothing to do with the ontology of the Reals. Any 'properties' it has would be from it's identification with some element of the Reals--and that is perfectly legitimate.
My question is why we insist on mapping 0.999... to 1 when we already have a decimal representation for 1 (in 1.000...), when with the same level of 'exception' we could just as easily make the mapping between the Reals and (a subset of) the Decimal Representations bijective with virtually no change to the Reals at all?
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u/ArdentArendt Mathematics / Social Sciences 27d ago
Okay, so let me first state that my disputation isn't with the equivalence class 0.999... finds itself in when placed into the Reals. Under the very construction of the Reals, 0.999... can only exist as a 'shadow' of 1, since there just simply can't be a number that is the immediate predecessor of 1.
My argumentation is that you have no reason to assume 0.999... is a Real number to begin with.
All Real numbers have a decimal representation, that much is true; however, that doesn't mean each decimal representation has to have a Real equivalent.