Ordinals - set of r-v'd functions on any interval in R and cardinality

Also, the theory of Borel equivalence relations is an important topic in set theory and has many applications in descriptive set theory and ergodic theory, among other areas.
  • #1
SiddharthM
176
0
just a cool fact I thought I'd share with anyone who's interested:

The set of real values functions on any interval in R has cardinality at least 2^c.

Pf: Consider characteristic functions defined on the interval, (a,b). (Note: a characteristic function is a function that can be defined on ANY domain and has range {0,1})

Let E be a subset of (a,b), then the characteristic, g(x) function of E over (a,b) i.e.

0 if x is not in E
g(x) =
1 if x is in E

Now for each subset E of (a,b) there corresponds a unique characteristic function defined on (a,b). Hence the set of all subsets of (a,b) and the set of characteristic functions defined on (a,b) are equivalent.
 
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  • #2
SiddharthM said:
The set of real values functions on any interval in R has cardinality at least 2^c.

that is clear - the set of functions from R to R has cardinality c^c, in fact, since it is, as a set, R^R.


Hence the set of all subsets of (a,b) and the set of characteristic functions defined on (a,b) are equivalent.

Yes, that is a fact everyone learns in their first meeting with sets and cardinality - it didn't need to be proved.
 
  • #3
c^c? what does that mean compared to 2^c? (2^alephnull)^(2^alephnull)?

I've met a lot of people who come out their first analysis course without even knowing what a characteristic function is. Surely these people study sets/cardinality (naively) in these courses? Outside of set theory classes where are cardinal #'s greater than c discussed?
 
  • #4
double-post sorry.
 
  • #5
c = 2^N

c^c = (2^N)^(c) = 2^(N * c) = 2^c
 
  • #6
SiddharthM said:
c^c? what does that mean compared to 2^c? (2^alephnull)^(2^alephnull)?

I've met a lot of people who come out their first analysis course without even knowing what a characteristic function is. Surely these people study sets/cardinality (naively) in these courses? Outside of set theory classes where are cardinal #'s greater than c discussed?
advanced set theory... (-:
but i guess you can meet this in logic and also in infinite combinatorics which is a new field.
i guess that also in non standard analysis, cause there we already constrcut the line of hyperreal numbers.
btw from what hurkyl gave you you need ofcourse to prove that for every infinite cardinality: a+a=a and a*a=a for that you need zorn's lemma, from this you can easiliy conclude from cantor bernstein theorem that for a>=b we have a+b=a and a*b=a.
 
  • #7
Here's an exercise for you: What's the cardinality of the set of real-valued continuous functions on R?
 
  • #8
SiddharthM said:
Outside of set theory classes where are cardinal #'s greater than c discussed?
In addition to loop quantum gravity's suggestions, topology (and in particular set theoretic topology) comes to mind. There are ways of using the order on an ordinal to induce a topology. Ordinals turn out to be very useful in giving examples of certain topological constructions and counterexamples to conjectures. They also pop up in curious places (thanks to the well ordering theorem and its many manifestations), e.g. one can prove that R^3 can partitioned into a union of disjoint unit discs using an ordinal-related argument.
 

1. What is the definition of ordinals in mathematics?

Ordinals are a type of number in mathematics that represent the order or position of an element in a sequence. They are typically used to describe the size or magnitude of a set, and are often denoted by a number followed by a superscripted letter (e.g. 1st, 2nd, 3rd, etc.)

2. How are ordinals and cardinals related?

Ordinals and cardinals are two types of numbers that describe the size or magnitude of a set. While ordinals represent the order or position of an element in a sequence, cardinals represent the actual quantity or number of elements in a set. In general, cardinals are used to count and compare sets, while ordinals are used to describe the order or arrangement of elements within a set.

3. What is the significance of the set of r-v'd functions on any interval in R?

The set of r-v'd functions on any interval in R is significant because it represents a collection of functions that map real numbers to real numbers. This set is important in mathematics because it allows for the study of continuous functions, which have many practical applications in fields such as physics, engineering, and economics.

4. What is the cardinality of the set of r-v'd functions on any interval in R?

The cardinality of the set of r-v'd functions on any interval in R is equal to the cardinality of the real numbers. This means that there are an uncountable number of r-v'd functions on any given interval in R, making this set a very large and diverse collection of functions.

5. How are ordinals and cardinals used in set theory?

Ordinals and cardinals play a crucial role in set theory, which is the branch of mathematics that deals with the properties and relationships of sets. In set theory, ordinals are used to describe the order or arrangement of elements within a set, while cardinals are used to compare the sizes or quantities of sets. These concepts are essential in understanding the structure and behavior of sets, which are fundamental objects in mathematics.

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