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Old 11 Apr 2012, 08:21 PM   #353859 / #26
David B
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All this reminds me that I have recorded the last of a BBC series of documentaries on maths, on unsolved problems, on my satellite box HD.

I wonder if the whole series is on any legit web site.

If anyone can find it, please post link.

David
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Old 12 Apr 2012, 08:54 PM   #354196 / #27
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Some things have developed so rapidly in the last sixty or so years that an item is out of date almost before you can publish your results. And the same is true of equipment: please to excuse the number of times I say "I think" below. I can remember the last half a dozen or so steps clearly enough but without my lab notebooks earlier details become vague.

Many years ago (late 1950s) I constructed pulse counting systems using - well call them electronic bottles - known as Dekatrons. The first variety involved a glow discharge which moved around a series of electrodes which ran around the circumference of a circular tube (low pressure neon I think). There was a later version which involved glowing numerals but I cannot remember whether with these the counting was done externally - I think not 'cos I'm sure I'd remember soldering up counting circuit with simple trannies or bottles).

Then some of the early minicomputers were programmed in Octal rather than hexadecimal: I think it was the PDP8 & PDP11 from DEC (?) & I think these were built using SSI circuits which involved three gates on a chip: but whether the machines themselves used binary or octal numbers I never knew.

And I also remember, about that time, arguments in favour of counting to base 60 or even 120: because these numbers were divisible by 1, 2, 3, 4, 5, 10, 15, 20.

Just fashion: some things change faster than hemlines.
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Old 13 Apr 2012, 05:03 PM   #354461 / #28
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ENIAC - GHN: IEEE Global History Network -- links to File:0513 Eniac decade ring counter.jpg - GHN: IEEE Global History Network
ENIAC used base 10, and it internally represented each digit with a set of 10 bits, one for each decimal digit.
0 = 10000 00000
1 = 01000 00000
...
9 = 00000 00001
A (Wikipedia)Dekatron looks like it implements ENIAC's approach in a single vacuum tube.

The University of Pennsylvania celebrated ENIAC's 50th anniversary by designing a computer chip that implements what ENIAC did:
Eniac-on-a-Chip Project
ENIAC-on-a-Chip - Penn Printout, Mar 96

As to base 60 or 120, how would they have been implemented? The ENIAC approach seems cumbersome.

Some computer hardware has been designed to do decimal arithmetic, like IBM's System/360 and its successors: System/370, S/390, z-series. It encodes decimal digits with binary-coded decimal: 0000 to 1001. Each byte contains 2 BCD digits.


As to octal vs. hexadecimal, those are mainly ways of displaying bits. Doing it as 1's and 0's can be awkward. Octal displays them in groups of 3, hexadecimal in groups of 4.

Wikipedia has articles on processors with these numbers of bits: 1, 4, 8, 12, 16, 18, 24, 31, 32, 36, 48, 60, 64, 128.

These numbers are word sizes rather than byte sizes, numbers of bits worked with as a group to represent integers and addresses and the like. Non-power-of-2 word sizes had been common among mainframes and minicomputers, but after about 1990, they were gone. Chip CPU's have almost universally used power-of-2 word sizes.
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Old 13 Apr 2012, 08:40 PM   #354518 / #29
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Now for the question of how many of various kinds of numbers there are. That question was first addressed by Georg Ferdinand Ludwig Philipp Cantor in his founding of set theory in 1874 - 1884.

He developed a theory of the cardinal numbers of sets, that being how many members/elements they contain. The infinite ones he called aleph-0, aleph-1, etc. He also worked on a theory of infinite ordinals, numbers in sequence, starting with one that he called omega. Unlike the finite case, infinite ordinals do not match onto infinite cardinals.

Two sets have the same cardinality / number of elements if their elements can be placed in one-to-one correspondence, if every element of each one can be matched with an element of the other one. Infinite sets have a very odd property. Unlike finite sets, an infinite set and a proper subset of it can have the same cardinality.

aleph-0 is the cardinality of the natural numbers; any set with that cardinality is called "countably infinite" or "countable", and any set with more than that cardinality is called "uncountable".

So let's see what's countable and what's not among subsets of the real numbers.
Positive integers: countable
Integers: countable
Rational numbers: countable
Real algebraic numbers: countable
Real computable numbers: countable
Real definable numbers: countable
Real numbers: uncountable -- their cardinality is not aleph-0, but a greater cardinality called C, for continuum
So there are infinitely many more real numbers than the others I've listed.

All these results has counterparts among the complex numbers, and indeed, among finite-length lists of real numbers.


What is C? Cantor showed that C > aleph-0, and he also showed that C is the cardinality of the set of all subsets of a countable set. That can be done with binary representations of real numbers between 0 and 1, where each digit's value is whether its location is a member of the corresponding set. 1 = true, 0 = false.

The set of all subsets of a set, including itself and the empty set, is the power set of a set. Cantor showed that a set with cardinality N has a power set with cardinality 2^N, and that 2^N > N for both finite and infinite N.

The sequence of power-set cardinalities starting at the countable cardinality is sometimes called the beth numbers: beth-0 = aleph-0, beth-1 = C, beth-(n+1) = 2^(beth-n)

But how are the alephs and the beths related? Does beth-1 = aleph-1 or some greater aleph? beth-1 = aleph-1 is the continuum hypothesis, and Cantor struggled with it. It's now known to be undecidable in the Zermelo-Fraenkel axioms of set theory, which means that there may be no way of telling.
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Old 17 Apr 2012, 07:18 PM   #355601 / #30
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So we have

Positive integers < Nonnegative integers < Integers < Rational numbers < Real algebraic numbers < Real computable numbers < Real definable numbers (all countable) < Real numbers (uncountable)

Complex numbers can be interpreted as pairs of real numbers that satisfy:
a = (ar,ai), b = (br,bi)
a+b = (ar+br, ai+bi), a*b = (ar*br-ai*bi, ar*bi+ai*br)

Polynomials can be interpreted as lists of real numbers:
a2*x2 + a1*x + a0 -> (a0,a1,a2) extended rightward with zeros as necessary.

a+b = (a0+b0, a1+b1, a2+b2, ...), a*b = (a0*b0, a1*b0+a0*b1, a2*b0+a1*b1+a0*b2, ...)

and likewise with multiple variables. It's possible to have lists with multiple indexing, of course. A matrix is a list with two indices that form a rectangular pattern:
((a00, a01), (a10, a11), (a20, a21))

A tensor is a multidimensional sort of of rectangular-pattern list. Vector = 1-index tensor, matrix = 2-index tensor, etc.


Vectors have a simple geometrical interpretation: points in space. Representing a point by a set of coordinate values was first explicitly stated by René Descartes, though it was hinted at by mathematicians in previous centuries. Vectors as an abstract mathematical object date back to the 19th cy., however. Matrices and tensors date back to the middle to late 19th cy. Albert Einstein helped popularize tensors through his use of them in general relativity; he wrote to a colleague who helped him out on tensor methods:
Quote:
I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.
That can be said about mathematical abstractions in general: it may take a lot of effort to learn some abstraction, but once one learns it, it can make one's life a Hades of a lot easier.

To illustrate such abstraction, let's consider what each vertex of a 3D model may get in 3D graphics:

Position: x, y, z
Normal (perpendicular) direction: x, y, z
Color: red, green, blue, opacity
Texture: x on image, y on image
So one's created a vector with 3 + 3 + 4 + 2 = 12 dimensions

Vectors live in "vector spaces", and these can be different from physical space. Color values can be interpreted as vectors in a vector space of colors, for instance. That's why RGB and CMYK and the like are sometimes called "color spaces". Vector spaces, like vectors themselves, are an abstraction.
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Old 18 Apr 2012, 05:54 AM   #355770 / #31
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A vector's components can be read off as component #1, component #2, component #3, etc. Or more compactly, for vector v,
v(1), v(2), v(3), etc.

Thus, vectors are much like functions, and functions have function spaces that are much like vector spaces. The cardinality of the set of real-number functions of real numbers is beth-2, though the set of continuous ones has cardinality beth-1, like the real numbers themselves.

So we have interpretations of beth-0, beth-1, and beth-2 outside of their definitions, but nothing comparable for any higher beth numbers. I recall Isaac Asimov once pointed out that one can't count beyond 3 in infinite sets, which refers to that.

-

Back to complex numbers, it turns out that they are all that's needed to solve real-coefficient polynomial equations. They also solve complex-coefficient ones, which can easily be turned into real-coefficient ones. They are thus algebraically closed. Complex algebraic numbers are also algebraically closed, as are complex computable and definable ones.

But some mathematicians have developed various generalizations of the complex numbers.

In 1843, William Rowan Hamilton considered additional imaginary-like numbers, and he worked out the "quaternions", defined with units i, j, k satisfying
i2 = j2 = k2 = i*j*k = -1

One can show from this definition that their multiplication is non-commutative:
i*j = k, j*i = -k, j*k = i, k*j = -i, k*i = j, i*k = -j
though it is still associative.

Quaternions had often been used to represent 3D vectors in the late 19th cy., but they've gone out of style for that purpose.

3D rotations can have their parameters expressed in quaternion form, with combinations of them being found from quaternion multiplication. Quaternions are often used in 3D computer graphics, like for smoothly rotating a (virtual) camera. Quaternions also appear in quantum mechanics as the Pauli matrices, introduced by Wolfgang Pauli in 1925 for the spins of spin-1/2 particles.

-

One can go further, to "octonions", and "sedenions", and the like, and the (Wikipedia)Cayley-Dickson construction will do that:

Real numbers -> Complex numbers -> Quaternions -> Octonions -> Sedenions -> ...

From a previous type of numbers, define pairs of them with these properties:

a = (ar,ai), b = (br,bi)
Conjugation: cjg(a) = (cjg(ar), - ai)
Real part: (a+cjg(a))/2
Imaginary part: (a-cjg(a))/2
Addition: a + b = (ar+br, ai+bi)
Multiplication: a*b = (ar*br - cjg(bi)*ai, bi*ar + ai*cjg(br))
The pairs generalize real and imaginary parts.
For real numbers, cjg(a) = a

One can define a norm:
norm(a) = a*cjg(a) = cjg(a)*a = real(a)2 + imag(a)2

As one advances in the construction, one loses multiplication properties, though it eventually levels off.
  • Real numbers: starting point
  • Complex numbers: loses ordering
  • Quaternions: loses commutativity: in general, a*b != b*a
  • Octonions: loses associativity: in general a*(b*c) != (a*b)*c, though it is still true if two of the a,b,c are equal: alternativity
  • Sedenions: loses alternativity, multiplication norming, and multiplication zero factoring: in general, norm(a*b) != norm(a)*norm(b), and there are some pairs of nonzero a,b where a*b = 0
  • Higher 2n-ions: like sedenions
However, power-associativity remains: am*an = am+n. In general,
an = P(n,real(a),imag(a)2) + Q(n,real(a),imag(a)2)*imag(a)

where one can use the complex numbers to find the P's and Q's:
(a + b*i)n = P(n,a,b2) + i*Q(n,a,b2)*b

John Baez has a nice page on this construction, though a rather technical one: Octonions
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Old 20 Apr 2012, 10:52 PM   #356997 / #32
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I'm venturing into this section of the cafe so as to learn more. I want to know why some numbers are called 'beautiful numbers'? I may have the term beautiful wrong but I over heard a conversation alluding to numbers that were called beautiful.
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Old 20 Apr 2012, 10:55 PM   #357000 / #33
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Fractional dimensions?

David
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Old 21 Apr 2012, 01:11 AM   #357041 / #34
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Rie, because they have some nice properties? Please give me a link.

David B, about fractional dimensions, are you referring to a fractal?

Imagine a system that's self-similar, that at length scale L that it is composed of N versions of itself at length scale L/n. Then its number of dimensions D is given by

N = nD

Some elementary examples.

Point: #, #, #, ...
1 = 10, 1 = 20, 1 = 30, ...
0 dimensions

Line: #, ##, ###, ...
1 = 11, 2 = 21, 3 = 31, ...
1 dimension

Surface:
#

##
##

###
###
###

1 = 12, 4 = 22, 9 = 32, ...
2 dimensions

Space:
#

## ##
## ##

### ### ###
### ### ###
### ### ###

1 = 13, 8 = 23, 27 = 33, ...
3 dimensions

Now for some fractal objects. Their numbers of dimensions from their self-similarity:
Cantor dust: log(2)/log(3) ~ 0.631
The Koch snowflake: log(4)/log(3) ~ 1.262
The Sierpinski carpet: log(8)/log(3) ~ 1.893
The Menger sponge: log(20)/log(3) ~ 2.727
(Wikipedia)List of fractals by Hausdorff dimension
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Old 24 Apr 2012, 05:04 AM   #358187 / #35
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There are 10 types of people in the world: those who understand binary and those who do not.
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Old 26 Apr 2012, 12:20 AM   #358894 / #36
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IPetrich I wish I could recall where I came across this this concept of a 'beautiful number'.. There is a web site from Peking university that explores the concept IPetrich, and it's very interesting. The number 2527 seems to be one example.

I can only direct you to this site. My first encounter of a 'beautiful' number came from a friend who mentioned it in conversation. And apparently as with gravity there's more to learn. I'll keep digging and let you know.

For me the symmetry of certain numbers is that they sort of harmonise in my brain... say 7 11 13.
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Old 26 Apr 2012, 12:27 AM   #358895 / #37
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The website has an administrator reachable by the address pojadmin@gmail.com.. Maybe this will help. Hope I'm not wasting anyone's time. but numbers are fascinating.
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Old 26 Apr 2012, 12:37 AM   #358896 / #38
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And another web site called 'Sciencetext' explores the BEAUTIFUL NUMBER 23. It references the Riemann Hypothosis... and another The Clay Mathematics Institution ( I may have got the last title wrong..)
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Old 26 Apr 2012, 07:15 AM   #358952 / #39
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Rie, you'd have a better idea of what you are looking for than I ever would, so I suggest that you do the e-mailing yourself. If there is anything too difficult, then please ask Daughter #2 or someone else with technical ability that you can meet in person or have a text-chat / instant-message conversation with.
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Old 26 Apr 2012, 08:26 AM   #358974 / #40
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I have a question. I assert that it is impossible to represent pi's magnitude in decimal. Right or wrong?

It's possible to imagine the infinite decimal digits as a series, but to get away from 'series' to magnitude, you have to collapse the concept away from it being a series (and therfore any integer base necessarily has to fall by the wayside if you want to represent the value itself)
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Old 26 Apr 2012, 08:41 AM   #358979 / #41
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I was just dipping my toes here. daughter is not necessary as the websites referencing 'beautiful' numbers are there if you google them. and i'm none the wiser re my numbers ... 3 7 11 except to say that I feel there's a pattern there.
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Old 26 Apr 2012, 08:46 AM   #358981 / #42
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all three numbers are prime. And are related by a multiple of four. Is that what you're referring to?
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Old 26 Apr 2012, 02:08 PM   #359027 / #43
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Quote:
Originally Posted by Rie View Post
I was just dipping my toes here. daughter is not necessary as the websites referencing 'beautiful' numbers are there if you google them. and i'm none the wiser re my numbers ... 3 7 11 except to say that I feel there's a pattern there.
Rie, why don't you give us some of them? Like what you consider some typical ones.

Would you need help in posting links?
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Old 26 Apr 2012, 10:21 PM   #359152 / #44
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Quote:
Originally Posted by Valheru View Post
I have a question. I assert that it is impossible to represent pi's magnitude in decimal. Right or wrong?
As a finite-length one, it cannot be done. It's easy to prove that every finite-length decimal representation is of a rational number, and it can also be proved that pi is irrational. One can easily show from those two propositions that pi lacks a FLDR.

If (N has a FLDR) then (N is rational)
is equivalent to
If (N is irrational) then (N does not have a FLDR)

It is not equivalent to
If (N is rational) then (N has a FLDR)
and one can easily find counterexamples, like 1/3 = 0.3333....

The counterexamples have an infinitely-repeating part: (top part)(repeating part)(repeating part)(repeating part)...

It's possible to show that every infinitely-repeating decimal representation is of a rational number, and it's also possible to show that every rational number has an IRDR. It's easy to show the first part, though the second part is more difficult. One can show it with the help of (Wikipedia)Euler's theorem, with a proof at Euler Function and Theorem and Fermat's Little Theorem. Euler's theorem will even supply an upper limit of the number of digits of the repeating part.

BTW, these decimal-representation discussions work for any number base.
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Old 27 Apr 2012, 12:47 AM   #359170 / #45
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Quote:
Originally Posted by Valheru View Post
It's possible to imagine the infinite decimal digits as a series, but to get away from 'series' to magnitude, you have to collapse the concept away from it being a series (and therfore any integer base necessarily has to fall by the wayside if you want to represent the value itself)
That conundrum is why mathematicians have come up with (Wikipedia)Limit (mathematics).

Here, one wants (Wikipedia)Limit of a sequence and (Wikipedia)Cauchy sequence.

Adding decimal-representation digits can be interpreted as creating a sequence, and its limit is for an infinite number of digits.

Imagine a sequence A = {a(1), a(2), a(3), ...}

It will have a limit a if for every e > 0, there is some N such that
for all n > N, |a(n) - a| < e

But one may not be sure that a limit is well-defined. So can one define convergence without using a limit? Yes one can, with a Cauchy sequence. A sequence A is one if for every e > 0, there is some N such that
for all m, n > N, |a(m) - a(n)| < e

Adding decimal digits will always make a Cauchy sequence, something which is easy to show.

Thus, {0, 0.3, 0.33, 0.333, 0.3333, ...} has limit 1/3
and {1, 1.4, 1.41, 1.414, 1.4142, ...} has limit sqrt(2)
and {3, 3.1, 3.14, 3.141, 3.1415, ...} has limit pi

Though 1/3 is a rational number, it is not a rational number with a power-of-10 denominator, contrary to its decimal-digits Cauchy sequence.

One can define arithmetic operations on Cauchy sequences. Doing it member-by-member creates a new Cauchy sequence, except for division by zero, of course. This shows that one can extend these operations to the sequences' limits.

In fact, one can define real numbers as the limits of Cauchy sequences of rational numbers.
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Old 28 Apr 2012, 11:38 PM   #359702 / #46
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Now I'm sticking my head out! I love numbers in threes... say, 555 or 333 etc. but those special numbers of 3 7 11 have always beckoned to me.

I wish like crazy that I was back to the classroom and understood algebraic solutions. And trigonometry and sine and cosine... but the powers that were then, that of being pushed into pure Academia and Literature instead of my love of 'scientific' subjects , won and that was only because it was reckoned that my chances of getting a scholarship that would pay all school expenses were better if I just stuck to Literature.
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Old 29 Apr 2012, 04:18 AM   #359747 / #47
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Maybe Daughter #2 could help you with that, Rie.


If one wants to get numbers from mathematical first principles, one can start with a set of axioms proposed by Italian mathematician Giuseppe Peano in 1889.

(Wikipedia)Peano axioms includes some axioms of equality, how one defines "equality". For relationship = and some set S,
  1. For all a in S, a = b
  2. For all a,b in S, a = b -> b = a
  3. For all a,b,c in S, a = b and b = c -> a = c
  4. For all a in S, a = b -> b in S
Now to the "natural numbers" N, defined as either nonnegative or positive integers. I'll do the nonnegative ones, since one can have a count of zero of something.
  • 0 is in N
  • For all a in N, S(x) is also in N -- S(a) is the successor function
  • There is no a in N such that S(a) = 0 -- the first one
  • For all a,b in N, S(a) = S(b) -> a = b -- S works backwards as well as forwards
  • For some condition P(a) defined for all a in N,
    if P(0) is true,
    and P(a) -> P(S(a)) for all a in N,
    then P(a) is true for all a in N
    -- mathematical induction
    The condition here is also called a "predicate", and this axiom can be stated with set membership rather than with a condition.

One can define addition with these axioms:
a + 0 = a
a + S(b) = S(a + b)

One can prove: a+b = b+a, a+(b+c) = (a+b)+c

Also multiplication:
a*0 = 0
a*S(b) = (a*b) + a

One can prove: a*b = b*a, a*(b*c) = (a*b)*a, a*(b+c) = (a*b)+(a*c)

Even ordering:
a <= b
means that there is some c in N such that b = a+c
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