What is Transfinite: Definition and 21 Discussions
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". Nevertheless, the term "transfinite" also remains in use.
Is it possible to calculate this :
Suppose the iterative root of ##2^x## :
##\phi(\phi(x))=2^x## (I suppose the Kneser calculation should work, it affirms that there is a real analytic solution)
Then how to compute ##\phi(\aleph_0)## ? (We know that ##2^{\aleph_0}=\aleph_1##).
Could this be...
Harold Simmons defined a simple but powerful notation for transfinite ordinals described in several articles available at
http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/ordinal-notations.html .
In summary :
- It uses lambda calculus formalism
- Fix f \zeta = f^\omega (\zeta+1) = limit...
I will first summarize the construction of ordinal numbers and introduce the definition of the binary Veblen function and of the notion of fundamental sequence.
Ordinal numbers start with natural numbers 0, 1, 2, 3, ... which are followed by ## \omega ## which represents the "simple" infinity...
Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane.
I want an extension of the domain of exp(⋅) and of h(⋅) so that
we can find values of these functions for x = Aleph(0).
My statement:
The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true...
Omega is the first transinite ordinal in the set of 0, 1, 2, ..., , , , ...
This set is well ordered, so is after therefore is before .
What is before ?
Options:
1. -1
This is confusing, since if is the first transfinite ordinal, then -1 should be the last finite ordinal, which simply...
So transfinite is larger than infinity right?
So if there was an infinitely large object, would a transfinite object be larger than the infinitely large object?
It is known that the Goodstein theorem
http://en.wikipedia.org/wiki/Goodstein's_theorem
which is a theorem about natural numbers, cannot be proved from the standard axioms of natural numbers, that is Peano axioms http://en.wikipedia.org/wiki/Peano_axioms .
It is also known that Goodstein...
Hi everybody,
In Enderton's "elements of set theory", he first discusses the red and then after some explanations
he discusses the brown as the general form of transfinite recursion theorem schema.
Then in the blue example, he uses the general form(brown) to show that the first form(red)is a...
I'm a bit confused on the subject of transfinite induction. I've read that it is equivalent to usual induction on the ordinal number ω (presumably a proof of this can be found in a standard book on the subject - any suggestions?)
Does anyone have an example of a case in which this isn't true...
Greetings, comrades!
In a previous thread, a user articulated a common argument:
His analogy mapping knights to horses makes intuitive sense, but how can we apply this idea to two infinite sets of knights and horses? How can we treat finite and transfinite sets equal in that sense and...
I am trying to understand the proof of Zorn's lemma from the axiom of choice, but I do not entirely understand the step where we create a increasing sequence (a_i) of sets in a partially ordered set S indexed by ordinals. It is defined through transfinite recursion, but how does that work...
Has anyone ever developed any sort of math involving donuts with an infinite number of holes? By donut, I mean a two-dimensional closed surface, curved in 3-space, with one 'hole'. Are there any results, of any kind, for 2-D donuts in 3-D space, with infinite number of holes?
Homework Statement
Show that if X is a subset of a well-ordered set (A, ≤ ) such that x0 > x1 > x2... then X must be finite.
The Attempt at a Solution
It seems like there's an obvious solution in that we know X must be well ordered, so has a least element. But by the question, X has a...
Let A,B,C be infinite sets. Define A^C as the set of all functions from C to A. Prove that if |A^C |=|B^C |, then |A|=|B|.
So I assume |A|<|B|. Since |A^C |<=|B^C | is true (proved theorem), I need only show that |A^C | not=|B^C |. Assume |A^C |=|B^C |, then there is a bijection f:A^C ->...
Let B be a basis for a vector space V (with an uncountable dimension!) over a field F, and let K be a linearly independent subset of V. Prove that there exists a subset S of B such that K U S is a basis for V.
I had to struggle a bit for this one. But I think I got it.
My recent interest in using transfinite induction in linear algebra has led me to pose a new question. (I will use c for the subset symbol)
Question: Use transfinite induction (not Zorn's lemma) to prove that if I is a linearly independent set and G is a set of generators (a spanning set)...
Hi,
First, please look at this example (It takes about 1 minute to load it) :
http://www.geocities.com/complementarytheory/PTree.pdf
From this example we can understand that if aleph0 is related to all N members then any n of n^aleph0 cannot be but 0.
The reason is very simple...