Questions about the logic of infinity

logics

I'm maintaining that:

- calling an integer a number is not an arbitrary act: (baz. is meaning-less, like 0, and if we call it a cod one might deduce it swims). We call it a number because a numeral has denoted a quantity for ages
- the properties of numerals, numbers and of operation(s) are not arbitrary, the object of definition, they are dictated by reality
- 0 cannot be considered a number because
a) does not denote a quantity, denotes a non-quantity, is meaning-less sign
b) its properties and properties of the operations involving 0 are different from numbers, or even arbitrary: left to the whim of the individual (0/0=54)
- if we are expressing principles, concepts in term of sets, we cannot include or adjoin alien elements.
can someone tell me if I'm wrong?

My question to micromass (who's an authorithy about sets) was: if we say that an entity is an element of a set, can we accept the fact that it is even slightly different ?
In the set fruit, we have different subsets for apples, pears and 'orange-lemmon etc' are together in a different subset of citrus, could we put them all together? Can we adjoin, just by definition, a dog to the set 'fruit', is this a 'lawful' logical or technical operation?

Mark44

Why all the random colors?
But micromass did NOT replace the word "integer" with a different word that has meaning. He used "bazalbieba", which I believe is a made-up word that is meaningless.
This is a very limited notion of what it means to be a number. Using this reasoning would you accept -3 as being a number? If so, would you also accept i (the imaginary unit) as being a number? How about omplex numbers? I certainly can't say that I now have 3 - 2i apples, but so what? If you can somehow accept complex numbers as being numbers, what about quaternions or octonions?

Mathematicians consider all of these to be numbers.
Yes. You seem to be thinking that numbers and operations are inseparable. They are not. Mathematicians have developed a number of structures, each of which involves a set of things (e.g., integers, real numbers, matrices, ...) and one or more operations (e.g. addition, multiplication, scalar multiplication).

Here are a few of these structures.
group - a set of things together with one operation.
ring - a set of things together with two binary operations, usually denoted by + and X.
integral domain - a ring in which multiplication satisfies a number of additional properties.
field - an integral domain in which every element except one (sometimes denoted z) is a unit.

It's not clear to me that you understand what being a member of a set entails. By definition, elements of a set are different from one another. From the wikipedia definition - "set - a collection of well defined and distinct objects." If the objects weren't distinct, we would have multiple copies of the same thing in the set. The objects in the set are distinct, so we can tell them apart. So a set is just a collection, which means you can put pretty much whatever you want to in a set.

For example, this is a perfectly valid set:
A = {apple, pear, lemon, orange, dog, cat}

Set A has a few obvious subsets:
{apple, pear, lemon, orange} - fruits
{apple, pear} - pomes
{lemon, orange} - citruses
{dog, cat} - mammals

SteveL27

That is not a useful definition of number.

First, a numeral is not a number.

Secondly, there is no general definition of "number" in math. The following can all be called numbers: the integers, the reals, the complex numbers, the p-adics, the integers mod 5, the transfinite ordinals, the transfinite cardinals, the Conway surreals, the Robinson nonstandard reals. That's not even an exhaustive list.

All of those can be called "numbers" within particular contexts. There is no general definition of numbers. Nor do numbers need to indicate quantities. If you imagine a number line, zero is just a point on the number line. It's like relabeling your address zero, the house to your left at -1, the house to your right as +1. It's just an arbitrary point on the line from which you begin defining left and right.

If numbers are a quantity, then do you regard the imaginary unit [itex]i[/itex] as a number? What quantity does it represent?

And what on earth can "ages" have to do with it? How long must a mathematical concept be accepted before you personally accept it?

nucl34rgg

It isn't enough to speak of "infinity." One must specify which infinity he/she is talking about. In other words, there are many different kinds of infinities.

(Here I will assume ZFC axioms.) For example, the cardinality of the real numbers is infinite, but it is also uncountable. The cardinality of the power set of the real numbers is also uncountable and infinite, but is greater than the cardinality of the reals. Also note that neither of these cardinal numbers are elements of the sets they describe. Another neat thing is that one can't consistently define such a thing as "the set of all cardinal numbers." Such a definition for a set leads to a paradox like the one described by Russel's Paradox.

Note that when one starts trying to describe "transfinite" numbers," it starts to become very important which axiomatic system one chooses to adopt. Certain axiomatic systems can lead to different conclusions.

If you are like me and are fascinated by these types of things, the area of math that deals with these types of questions is set theory.

Also, as SteveL27 has stated above, the concept of "number" itself has been difficult to define. It seems to be something primitive akin to that of "set" from which one must simply start.

logics

I am grateful to you all for your contributions, I'll reply in the thread: qualitative vs. quantitative . . My concern is not about the definition of number, but the logics of ∞. Could you tell me why the answer to that question should be no, if ∞ is not necessary?

just two notes:
- Steve, 'numeral' is a hyper[o]nym: a term that includes what you call now numbers (1,2...) , other numbers (I,V,X...) and their language 'word-form' (one, two,....)
- Mark₄₄, a set is indeed made up by different elements, but is identified by Cantor's 'sentential formula', [Edit ,i.e.: the rule for set membership : {x | x is a natural number/fruit/numeral...} ], the set 'numeral' has subsets identified by hyponyms: one may be 'glyph-numerals' and another can be 'letter/word-numerals' the subset 'glyphs', in its turn, has subsets ''Indian/Arabic-number/glyphs', 'Latin glyphs' etc . We surely cannot make a jumble.

Edit: If one wants to adjoin 0 to the set { x | x is a natural number} he must prove it's a number. In the FAQ 0 is not defined,
micromass hasn't replied directly to post#18 but in this post has just confirmed my thesis, if 'non-quantity' means 'not measurable'. Now we must only find out if we can adjoin 0 to N when we think 'it makes no sense to make 0 a number':

- Mark₄₄ you may make a jumble set if you list all its elements : {0, 1, 2 ,..} but I maintain you must change the rule: {x | x is a baz.albieba} [a baz. is either 0 or n !(more or less so)] }

I hope HallsofIvy will tell me where I am wrong!

SteveL27

New term to me, but I'm looking at this:

http://en.wikipedia.org/wiki/Hyponymy

and I do not believe that a number is a numeral. It's a very basic fact of mathematical culture that a numeral is not a number; a numeral is a representation of a number.

In hyponymy, we are talking about types. A rectangle is a shape. This is not the relationship between a numeral and a number. I wonder if you are not stretching the concept a bit. If there is some field of knowledge that considers a numeral a number, that's fine; but mathematics is not such a field of knowledge. A numeral is not a number. Of that I am certain.

The fact that you consider V and 5 to be different numbers is a clue that you are stretching the concept too far. V and 5 are different numerals representing the same number.

Mark44

I have no idea what you mean by "logics of ∞".
We surely can. Whether it suits a purpose is a different discussion. Again, you misunderstand what a set is, which is nothing more than a collection of things. A set can contain whatever you choose to put in it.

Mark44

I'm with Steve on this: the terms number and numeral are two different things. Numerals are the symbols that we use to represent a particular number.

logics

See here:numeral

logics

if wiki is not considered reliable, the ultimate authorithy on English language, SOED, II vol., p. 955:
Numeral, n,:
A: a word expressing a number [ one, two,...]
B: a figure [ 8 ..] or symbol [ V, β, ,,] or a group of these [346, MCMLIII....], denoting a number

I hope that's that !

Mark44

What's your point? Both definitions in the article you linked to makes a distinction between 'number' and 'numeral', which is what Steve and I have already said.

Mark44

Do you not understand what they are saying? They are NOT saying that a numeral IS a number. A numeral is a representation of a number.

logics

is 5 a number? so it is not a numeral
Next you say (colour added)
Steve, a number is the representation of a quantity (SOED) or whatever you choose, a representation of a representation makes little sense, V and 5 represent * 5 → 5 represents 5 ?
Five, V and 5 are three different numerals [one English word and two symbols (one Latin and one Indian-Arabic)] representing the same quantity [OOOOO]. I hope it is all clear, now.
Can you answer the question in post #22?, do you know how to copy this paragraph from wiki so we can quote it? You'll oblige me!

Thanks,