Irrational numbers and Physical constants

[aleph_1]

Dragons aren't reducible to perception, but infinite sets are. Reducible doesn't mean perceivable, it just means a concept can be connected to reality by figuring out how the concept was developed. There's nothing problematic about infinite sets, at least nothing problematic suggested by what was discussed about the nature of concepts.

To falsify your statement, one merely needs to consider the consequences. It is easy to define the Countable infinity. This concept applies to your mental existent called the Natural Numbers so it has immediate application. It is easy to construct the Real Numbers. Then it is simple to show that the real numbers are not countably infinite. Then it is simple, using Cantor's Theorem, to construct an infinity of infinities of different sizes. These all are valid concepts for the same reason you state. Their distinction is made possible through the mental constructs you advocate. The implication of your philosophy is the existence of an infinity of infinities of different sizes. Quod Erat demonstrandum.

[Eiuol]

The implication of your philosophy is the existence of an infinity of infinities of different sizes.

 

I'm fine with that implication and I don't think anyone has sufficiently demonstrated that this is a problem for Objectivist philosophy. Whether or not it is true is a mathematical issue, but it doesn't reveal a contradiction.

[aleph_1]

I'm fine with that implication and I don't think anyone has sufficiently demonstrated that this is a problem for Objectivist philosophy. Whether or not it is true is a mathematical issue, but it doesn't reveal a contradiction.

Your comments mean that accepting ideas that have no perceptual validation are acceptable. You believe in dragons! It is not contradictions that are at issue. There are many perfectly rational but false ideas. The issue is validation through reduction to perception. This you have not done!

You have permitted the introduction of meaninglessness into you system of concepts. That is not Objectivist!

[GrandMinnow]

 natural numbers go on forever, but certainly countable as n+1. If I give you a natural number, you'll always know what's next. 1000, 1001, etc.

 

Yes, you have the basic idea. The definition of 'countable' is "S is countable if and only if there is a 1-1 correspondence between S and the set of natural numbers or there is a natural number such that there is a 1-1 correspondence between S and said natural number." A set S if countably infinite if and only if there is a 1-1 correspondence between S and the set of natural numbers. So a set is countable if and only if the set is either finite or countably infinite. So it turns out that any countable set S can be ordered (if the set is countably infinite, then use any 1-1 correspondence with the set of natural numbers; and if the set is finite  use any 1-1 correspondence with a natural number) so that (1) S has a first member, and (2) if S is infinite, then for any member of S there is exactly one next member, and if S finite then then for any member other than the last member, there is exactly one next member, and (3) for any member x of S, by a finite number of steps, we "arrive" at x by starting with the first member of S and going stepwise to each next member until we "arrive" at x. In contrast, that is not possible with an uncountable set. 

 

I don't know if this applies to rational numbers

 

It does. The set of rational numbers is countable. Even though the standard ordering of the set of rational numbers is a dense ordering, there is a 1-1 correspondence (by "zig zagging" through numerators and denominators) between the set of rational numbers and the set of natural numbers, thus there are other orderings of the set of rational numbers that are not dense orderings but rather are isomorphic to the standard ordering of the set of natural numbers.

 

it wouldn't make sense for infinity to have varying sizes

 

We need to be careful in the terminology. In this set theory we're talking about, there is no such object that is called 'infinity'. Rather, 'is infinite' is an ADJECTIVE. Being infinite or not infinite is a PROPERTY of sets. Some sets are finite and some sets are infinite, but there is no object itself that we call "infinity" (well, in another context there is, but it's not with regard to cardinality, so let's set that other context aside for now). The definitions are:

 

S is finite if and only if there is a natural number that is in 1-1 correspondence with S.

 

S is infinite if and only if S is not finite.

 

Now, it turns out that there are sets that are infinite but not in 1-1 correspondence with each other, while one set (call it S) is in 1-1 correspondence with a proper subset of the other (call it T). When this obtains we say that T is of greater infinity (has greater infinite cardinality) than S. Again, it is not the case that there is an object called 'infinity', rather it is that there are sets having the property of being infinite and some of them have greater infinite cardinality than others.

Edited February 13, 2013 by GrandMinnow

[GrandMinnow]

Oists philosophy is inconsistent with the Infinity Axiom since that axiom implies the actual existence of an infinity.

 

It might be that Objectivism does not accept even the FRAMEWORK of a formalized (or even unformalized) language and mathematical logic in which a formal axiom of infinity is considered.

 

Also, as you know (so this comment is directed generally, not personally), by "actual infinity" in this context we don't mean "tangible" like a tree and not necessarily even platonically (though many mathematicians and philosophers of mathematics do ascribe to certain forms of realism (aka 'platonism')). Rather, by saying that the axiom of infinity entails the existence of a set that is actually infinite, we may mean as little as that there is (in whatever ontological sense one regards 'is', ranging from platonic through fictionalism through pure formalism, etc.) a set that includes as members all and only the natural numbers, and that we don't obligate ourselves to understand that notion in terms of mere potentiality. Recall, for example, the formalist Abraham Robinson who declares that mention of 'infinity' is meaningless yet he works unashamedly with ZFC (with its axiom of infinity). 

 

From [the notion of potential infinity] we may construct Natural Numbers, but not "the set of Natural Numbers". We may also construct Rational Numbers, but not the "set of Rational Numbers".

 

There may be approaches in which infinity is accepted only as potentiality but where we also formally construct rational numbers, but it's not obvious how to do that. Recall that in the standard approach, even a single rational number is an equivalence class of natural numbers (this equivalence class being an infinite set itself). 

Edited February 13, 2013 by GrandMinnow

[GrandMinnow]

is infinity a concept that refers to any pattern that goes on endlessly and also refers to the whole pattern at once?

In the context of this set theory we're talking about, there is nothing that is called 'infinity'. Rather there is the property 'is infinite'. If you overlook this distinction, then the set theoretic approach will always remain confusing to you.

 

Some sets have the property of being finite (we say these sets are finite). But there is no set that we call "Finity". And some sets don't have the property of being finite (we say these sets are infinite). And there is no set that we call "Infinity".

Edited February 13, 2013 by GrandMinnow

[Eiuol]

In the context of this set theory we're talking about, there is nothing that is called 'infinity'. Rather there is the property 'is infinite'. If you overlook this distinction, then the set theoretic approach will always remain confusing to you.

Yeah, you've pointed out imprecision in my word choice. Infinity as a property of sets makes a lot more sense to say (and perhaps as I suggested earlier infinity is a property of concepts in general, too). I'll have to think some more but you clarified a lot, thanks.

[aleph_1]

There may be approaches in which infinity is accepted only as potentiality but where we also formally construct rational numbers, but it's not obvious how to do that. Recall that in the standard approach, even a single rational number is an equivalence class of natural numbers (this equivalence class being an infinite set itself). 

I would like to thank GrandMinnow for demonstrating patience and erudition in the given responses. As a confession, I have not studied set theory for 28 years, so the rust is thick. Yes, GrandMinnow has compelled me to crack open my old text (Enderton) to check on the definition of rational numbers. It is true that rational numbers are defined as equivalence classes of ordered pairs of integers. These ordered pairs correspond to fractions, so individual fractions may be defined without the Infinity Axiom.

I am currious about your other concerns about the set theoretic foundations of math. May we at least claim that we have here uncovered serious issues relating to the notion of number and perceptual reduction? I think a paper could easily be written.... GrandMinnow?

[GrandMinnow]

the rust is thick

Same here.

 

Just to be clear, with, for example, Enderton (which is a typical treatment):

 

Integers are equivalence classes of natural numbers, so each integer is itself an infinite set. An ordered pair of natural numbers, such as <0 1> is finite, but the integer called "negative one" is an equivalence class that is an infinite set; <0 1> is one of the members.

 

Rationals are equivalence classes of integers, so each rational is itself an infinite set. And ordered pair of integers (a fraction), such as <integer_1 integer_2> is finite, but the rational number we could call "the ratio of the integer 1 and the integer 2" is an equivalence class that is an infinite set; <integer_1 integer_2> is one of the members.

 

I am curious about your other concerns about the set theoretic foundations of math. May we at least claim that we have here uncovered serious issues relating to the notion of number and perceptual reduction?

I feel that there may be some perplexing philosophical problems with set theory, but the notion of perceptual reduction doesn't happen to be very high among my own personal concerns.