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The All New Groundspeak Uk Pub Quiz!

The Golem
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The Mars Bars

+The Mars Bars

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Posted

OK while we are touching on planes, heres mine. A none googleable one methinks.

I once worked at Stanstead Airport and in late 86 early 87 (can't remember exactly) Rolls Royce went public.

RR had a day of RR engined planes throughout the company's history. Every one either landed or took off that day, except one. Now guessing the one would be too easy so I want the first plane that did a take off /flyby /landing and the one that didn't touch the Stanstead runway on that day.

 

Carlos

PS Confirmation ding will be in the morning cos I'm off to the land of nod :)

 

Spitfire, Concorde?

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Nediam

Nediam

Posted
Posted (edited)

Spitfire, Concorde?

 

Oi! I was gonna say them! :):):D

 

(was just about to reply when I noticed there was another page after the one I was on :) )

 

 

I'll stick with Concorde as it's most likely the one that didn't land. (If it's not I'll have a go with the Vulcan Bomber or a Harrier)

 

In case the Spitfire is wrong, I'll have a go with a Vickers

 

 

Edit:- I do have a great picture of a Harrier flying about 2 feet off the ground (don't know if it was photoshopped) which I would have posted but Photobucket.com is down for maintenance :huh: (sound familiar :P )

Edited by Nediam
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Carlos & The Birdie Crew

+Carlos & The Birdie Crew

Posted
Posted

I can clearly remember the opening flight which was (I promise) a gypsy tiger moth bi plane.

 

We sat and listened to the tower introduce the planes over the radio as each plane took off or landed. The bizarre thing was all the planes were reffered to by a flight number in communication with the pilots (where possible) except concord were the tower addressed the actual plane directly.

 

The Mars Bars got the answer I was expecting so I can only Ding him. Sorry if it's not as expected.

 

Regards, Carlos

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The Forester

+The Forester

Posted
Posted

All of the Moths had one variant ir another of the Dripsy engine. You really couldn't shoehorn any other type of engine into those airframes.

 

Anything other than an inverted straight four would either mean that the prop thrustline would be so low that using any significant amount of power would pitch the nose up to stalling point faster than the pilot could swear, or else the line of cylinders would totally block forward vision. I suppose you could fit a horizontally opposed flat, but then the forward visibilty and vis to the side of the nose would be non-existent due to the jugs getting in the way of the pilot's sightline. Taxying, takeoff and approaches would be an absolute pig.

 

No, I think they were right to fit only the lovely old Dripsy, even though she does leak as much oil on the ground as she burns when the engine is running!

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The Mars Bars

+The Mars Bars

Posted
Posted (edited)

It looks like one of the buildings in Cardiff that they seem to spend a long time standing on top of in Torchwood.

 

Nothing to do with Torchwood or Cardiff....

 

Another shot that will make things a little clearer...

 

3580.jpg

 

Not Cardiff... :ph34r:

 

Cheers

Dave

Edited by The Mars Bars
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Archer4

+Archer4

Posted
Posted

:)

 

Of what is the following a mathematical proof? :blink:

 

Peano's Postulates are:

1. Let S be a set such that for each element x of S there exists a

unique element x' of S.

2. There is an element in S, we shall call it 1, such that for every

element x of S, 1 is not equal to x'.

3. If x and y are elements of S such that x' = y', then x = y.

4. If M is any subset of S such that 1 is an element of M, and for

every element x of M, the element x' is also an element of M, then

M = S.

Just as a matter of notation, we write 1' = 2, 2' = 3, etc. We define

addition in S as follows:

(a1) x + 1 = x'

(a2) x + y' = (x + y)'

The element x + y is called the sum of x and y.

From (a1), with x = 1, we see that...

:blink:

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The Golem

+The Golem

Posted
  • Author
Posted (edited)

Oh - that old chestnut....

 

Peano axioms

In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. The axioms are usually encountered in a first-order form, where the crucial second-order induction axiom is replaced by an infinite first-order induction schema, and Peano Arithmetic (PA) is by convention the name of the widely used system of first-order arithmetic given using this first-order form. However, Peano arithmetic is essentially weaker than the second-order axiom system, since there are nonstandard models of Peano arithmetic, and the only model for the Peano axioms (considered as second-order statements) is the usual system of natural numbers (up to isomorphism).

 

Peano first gave his axioms in a Latin text Arithmetices principia, nova methodo exposita published in 1889 (Peano 1889), where Peano gave nine axioms, four axioms specifying the behaviour of the equality relation and five rules involving the specifically arithmetic terms for zero and successor. It is the latter five rules that are usually intended when one discusses the Peano axioms. Peano took logical principles to be given.

 

Peano arithmetic constitutes a fundamental formalism for arithmetic, and the Peano axioms can be used to construct many of the most important number systems and structures of modern mathematics. Peano arithmetic raises a number of metamathematical and philosophical issues, primarily involving questions of consistency and completeness.

 

The axioms

Informally, the Peano axioms may be stated as follows:

 

0 is a natural number.

Every natural number a has a successor, denoted by Sa or a'.

No natural number has 0 as its successor.

Distinct natural numbers have distinct successors: a = b if and only if Sa = Sb.

If a property holds for 0, and holds for the successor of every natural number for which it holds, then the property holds for all natural numbers. (This axiom of induction is also called "mathematical induction". Its point is to limit the natural numbers to just those which are required by the other axioms.)

Peano's original axioms (1889) are preceded with the definitions:

 

"The sign N means number (positive integer).

"The sign 1 means unity" (italics in original, van Heijenoort (1976) p. 94)

His first axiom is "1 ε N" (ibid), the ε signifying "is an element/member of". No mention is made of the sign "0" (also cf commentary by van Heijenoort p. 83, and Dedekind's Letter to Keferstein" (1890) p.100).

 

Letting the first natural number be 1 requires replacing 0 with 1 in the above axioms. Starting with 1 changes the recursive definitions of addition and multiplication given below, if one wishes the symbol "1" to represent what is normally considered the number "1". In the representation of unary numbers on the tape of a Turing machine, and in particular the Post-Turing machine, the place-holder "0" is frequently represented by a single tally mark " | ", often written as " 1 ", the unit by two marks " | | ", etc. (cf Davis (1974) p. 72).

 

More formally and following Dedekind (1888), define a Dedekind-Peano structure as an ordered triple (X, x, f), satisfying the following properties:

 

X is a set, x is an element of X, and f is a map from X to itself.

x is not in the range of f.

f is injective.

If A is a subset of X satisfying:

x is in A, and

If a is in A, then f(a) is in A

then A = X.

 

The following diagram sums up the Peano axioms:

 

where each of the iterates f(x), f(f(x)), f(f(f(x))), ... of x under f are distinct.

 

Without the axiom of induction, one could have closed loops of any finite length or extra copies of the integers or natural numbers added to the usual natural numbers.

 

Binary operations and ordering

The above axioms can be augmented by definitions of addition and multiplication over the natural numbers N, and by the usual ordering of N.

 

To define addition '+' recursively in terms of successor and 0, let a+0 = a and a+Sb = S(a+:) for all a, b. This turns into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and is therefore embeddable in a group. The smallest group containing the natural numbers is the integers.

 

Given this definition of addition and 1 := S0, we see that: b+1 = b+S0 = S(b+0) = Sb, that is, b+1 is simply the successor of b.

 

Given successor, addition, and 0, define multiplication '·' by the recursive axioms a·0 = 0 and a·Sb = (a·:blink: + a. Hence is also a commutative monoid with identity element 1. Moreover, addition and multiplication are compatible by virtue of the distribution law:

 

a·(b+c) = (a·:blink: + (a·c).

Define the usual total order on N, ≤, as follows. For any two natural numbers a and b, a ≤ b if and only if there exists a natural number c such that a+c = b. This order is compatible with addition and multiplication in the following sense: if a, b and c are natural numbers and a ≤ b, then a+c ≤ b+c and a·c≤b·c. An important property of N is that it also constitutes a well-ordered set; every nonempty subset of N has a least element.

 

Peano arithmetic

Peano Arithmetic (PA) reformulates the Peano axioms as a first order theory with two binary operations, addition and multiplication, recursively defined as in the preceding section, and denoted by infix '+' and '·', respectively. The conceptual change is that the Axiom schema of Induction is replaced by a schema permitting induction only over arithmetical formulae φ, whose non-logical symbols are just 0, S, '+', '·', and lower case letters as variables ranging over the natural numbers. The new axiom schema represents a countably infinite set of axioms.

 

The axioms of PA are:

 

 

for any formula in the language of PA.

 

 

PA does not require the predicate "is a natural number" because the universe of discourse of PA is just the natural numbers N. While no explicit existential quantifiers appear in the above axioms, four tacit quantifiers of that nature follow from the closure of the natural numbers under zero, successor, addition, and multiplication; and there may be implicit existential quantifiers in the axioms of induction within the

 

Existence and uniqueness

A standard construction in set theory shows the existence of a Dedekind-Peano structure. First, we define the successor function; for any set a, the successor of a is S(a) := a ∪ {a}. A set A is an inductive set if it is closed under the successor function, i.e. whenever a is in A, S(a) is also in A. We now define:

 

0 := {}

N := the intersection of all inductive sets containing 0

S := the successor function restricted to N

The set N is the set of natural numbers; it is sometimes denoted by the Greek letter ω, especially in the context of studying ordinal numbers.

 

The axiom of infinity guarantees the existence of an inductive set, so the set N is well-defined. The natural number system (N, 0, S) can be shown to satisfy the Peano axioms. Each natural number is then equal to the set of natural numbers less than it, so that

 

0 := {}

1 := S(0) = {0}

2 := S(1) = {0,1} = {0, {0}}

3 := S(2) = {0,1,2} = {0, {0}, {0, {0}}}

and so on. This construction is due to John von Neumann.

 

This is not the only possible construction of a Dedekind-Peano structure. For instance, if we assume the construction of the set N = {0, 1, 2,...} and successor function S above, we could also define X := {5, 6, 7,...}, x := 5, and f := successor function restricted to X. Then this is also a Dedekind-Peano structure.

 

The lambda calculus gives another construction of the natural numbers that satisfies the Peano axioms.

 

Two Dedekind-Peano structures (X, x, f) and (Y, y, g) are said to be isomorphic if there is a bijection φ : X → Y such that φ(x) = y and φf = gφ. It can be shown that any two Dedekind-Peano structures are isomorphic; in this sense, there is a "unique" Dedekind-Peano structure satisfying the Peano axioms. (See the categorical discussion below.)

 

Categorical interpretation

The Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1 is the following category:

 

The objects of US1 are all ordered triples (X, x, f), where X is a set, x is an element of X, and f is a set map from X to itself.

For each (X, x, f), (Y, y, g) in US1, HomUS1((X, x, f), (Y, y, g)) = {set maps φ : X → Y | φ(x) = y and φf = gφ}

Composition of morphisms is the usual composition of set mappings.

The natural number system (N, 0, S) constructed above is an object in this category; it is called the natural unary system. It can be shown that the natural unary system is an initial object in US1, and so it is unique up to a unique isomorphism. This means that for any other object (X, x, f) in US1, there is a unique morphism φ : (N, 0, S) → (X, x, f). That is, that for any set X, any element x of X, and any set map f from X to itself, there is a unique set map φ : N → X such that φ(0) = x and φ(a + 1) = f(φ(a)) for all a in N. This is precisely the definition of simple recursion.

 

This concept can be generalised to arbitrary categories. Let C be a category with (unique) terminal object 1, and let US1© be the category of pointed unary systems in C; i.e. US1© is the following category:

 

The objects of US1© are all ordered triples (X, x, f), where X is an object of C, and x : 1 → X and f : X → X are morphisms in C.

For each (X, x, f), (Y, y, g) in US1©, HomUS1©((X, x, f), (Y, y, g)) = {φ : φ is in HomC(X, Y), φx = y, and φf = gφ}

Composition of morphisms is the composition of morphisms in C.

Then C is said to satisfy the Dedekind-Peano axiom if there exists an initial object in US1©. This initial object is called a natural number object in C. The simple recursion theorem is simply an expression of the fact that the natural number system (N, 0, S) is a natural number object in the category Set.

 

Metamathematical discussion

These axioms are given here in a second-order predicate calculus form. See first-order predicate calculus for a way to rephrase these axioms to be first-order.

 

Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers.

 

On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic.

 

If one replaces the last axiom with the schema:

 

If P(0) is true; and for all x, P(x) implies P(x + 1); then P(x) is true for all x.

for each first order property P(x) (an infinite number of axioms) then although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrarily large cardinality - by the compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization; by an "upward Löwenheim-Skolem theorem", there exist models of all cardinalities. Moreover, if Dedekind's proof that the second order Peano Axioms have a unique model is viewed as a proof in a first order axiomatization of set theory such as Zermelo–Fraenkel set theory, the proof only shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism; nonstandard models of set theory may contain nonstandard models of the second order Peano Axioms.

 

When the axioms were first proposed, people such as Bertrand Russell agreed these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent. If a proof can exist that starts from just these axioms, and derives a contradiction such as P AND (NOT P), then the axioms are inconsistent, and don't really define anything. David Hilbert posed a problem of proving consistency using only finite logic as one of the problems on his famous list.

 

But in 1931, Kurt Gödel in his celebrated second incompleteness theorem showed such a proof cannot be given in any subsystem of Peano arithmetic. Furthermore, we can never prove that any axiom system is consistent within the system itself, if it is at least as strong as Peano's axioms, instead one must prove the consistency of the system using a different axiomatic system. Although it is widely believed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this is not clear, and depends on exactly what one means by a finitistic proof: Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. As Gentzen has explained it himself: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers). Whether or not this really counts as the "finitistic proof" that Hilbert wanted is unclear: the main problem with deciding this question is that there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

 

Most mathematicians assume that Peano arithmetic is consistent, although this relies on either intuition or on accepting Gentzen's proof. However, early forms of naïve set theory also intuitively looked consistent, before the inconsistencies were discovered.

 

Founding a mathematical system upon axioms, such as the Peano axioms for natural numbers or axiomatic set theory or Euclidean geometry is sometimes labeled the axiomatic method or axiomatics.

 

 

Can we please please try and keep the questions at a level that might be found in a Wednesday night pub quiz.... :blink:

 

 

In the meantime....

 

Which German line lay opposite the French Maginot Line?

Edited by The Golem
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Nediam

Nediam

Posted
Posted (edited)

:)

 

Of what is the following a mathematical proof? :blink:

 

Peano's Postulates are:

1. Let S be a set such that for each element x of S there exists a

unique element x' of S.

2. There is an element in S, we shall call it 1, such that for every

element x of S, 1 is not equal to x'.

3. If x and y are elements of S such that x' = y', then x = y.

4. If M is any subset of S such that 1 is an element of M, and for

every element x of M, the element x' is also an element of M, then

M = S.

Just as a matter of notation, we write 1' = 2, 2' = 3, etc. We define

addition in S as follows:

(a1) x + 1 = x'

(a2) x + y' = (x + y)'

The element x + y is called the sum of x and y.

From (a1), with x = 1, we see that...

:)

 

Oh - that old chestnut....

 

Peano axioms

In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. The axioms are usually encountered in a first-order form, where the crucial second-order induction axiom is replaced by an infinite first-order induction schema, and Peano Arithmetic (PA) is by convention the name of the widely used system of first-order arithmetic given using this first-order form. However, Peano arithmetic is essentially weaker than the second-order axiom system, since there are nonstandard models of Peano arithmetic, and the only model for the Peano axioms (considered as second-order statements) is the usual system of natural numbers (up to isomorphism).

 

Peano first gave his axioms in a Latin text Arithmetices principia, nova methodo exposita published in 1889 (Peano 1889), where Peano gave nine axioms, four axioms specifying the behaviour of the equality relation and five rules involving the specifically arithmetic terms for zero and successor. It is the latter five rules that are usually intended when one discusses the Peano axioms. Peano took logical principles to be given.

 

Peano arithmetic constitutes a fundamental formalism for arithmetic, and the Peano axioms can be used to construct many of the most important number systems and structures of modern mathematics. Peano arithmetic raises a number of metamathematical and philosophical issues, primarily involving questions of consistency and completeness.

 

The axioms

Informally, the Peano axioms may be stated as follows:

 

0 is a natural number.

Every natural number a has a successor, denoted by Sa or a'.

No natural number has 0 as its successor.

Distinct natural numbers have distinct successors: a = b if and only if Sa = Sb.

If a property holds for 0, and holds for the successor of every natural number for which it holds, then the property holds for all natural numbers. (This axiom of induction is also called "mathematical induction". Its point is to limit the natural numbers to just those which are required by the other axioms.)

Peano's original axioms (1889) are preceded with the definitions:

 

"The sign N means number (positive integer).

"The sign 1 means unity" (italics in original, van Heijenoort (1976) p. 94)

His first axiom is "1 ε N" (ibid), the ε signifying "is an element/member of". No mention is made of the sign "0" (also cf commentary by van Heijenoort p. 83, and Dedekind's Letter to Keferstein" (1890) p.100).

 

Letting the first natural number be 1 requires replacing 0 with 1 in the above axioms. Starting with 1 changes the recursive definitions of addition and multiplication given below, if one wishes the symbol "1" to represent what is normally considered the number "1". In the representation of unary numbers on the tape of a Turing machine, and in particular the Post-Turing machine, the place-holder "0" is frequently represented by a single tally mark " | ", often written as " 1 ", the unit by two marks " | | ", etc. (cf Davis (1974) p. 72).

 

More formally and following Dedekind (1888), define a Dedekind-Peano structure as an ordered triple (X, x, f), satisfying the following properties:

 

X is a set, x is an element of X, and f is a map from X to itself.

x is not in the range of f.

f is injective.

If A is a subset of X satisfying:

x is in A, and

If a is in A, then f(a) is in A

then A = X.

 

The following diagram sums up the Peano axioms:

 

where each of the iterates f(x), f(f(x)), f(f(f(x))), ... of x under f are distinct.

 

Without the axiom of induction, one could have closed loops of any finite length or extra copies of the integers or natural numbers added to the usual natural numbers.

 

Binary operations and ordering

The above axioms can be augmented by definitions of addition and multiplication over the natural numbers N, and by the usual ordering of N.

 

To define addition '+' recursively in terms of successor and 0, let a+0 = a and a+Sb = S(a+B) for all a, b. This turns into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and is therefore embeddable in a group. The smallest group containing the natural numbers is the integers.

 

Given this definition of addition and 1 := S0, we see that: b+1 = b+S0 = S(b+0) = Sb, that is, b+1 is simply the successor of b.

 

Given successor, addition, and 0, define multiplication '·' by the recursive axioms a·0 = 0 and a·Sb = (a·B) + a. Hence is also a commutative monoid with identity element 1. Moreover, addition and multiplication are compatible by virtue of the distribution law:

 

a·(b+c) = (a·B) + (a·c).

Define the usual total order on N, ≤, as follows. For any two natural numbers a and b, a ≤ b if and only if there exists a natural number c such that a+c = b. This order is compatible with addition and multiplication in the following sense: if a, b and c are natural numbers and a ≤ b, then a+c ≤ b+c and a·c≤b·c. An important property of N is that it also constitutes a well-ordered set; every nonempty subset of N has a least element.

 

Peano arithmetic

Peano Arithmetic (PA) reformulates the Peano axioms as a first order theory with two binary operations, addition and multiplication, recursively defined as in the preceding section, and denoted by infix '+' and '·', respectively. The conceptual change is that the Axiom schema of Induction is replaced by a schema permitting induction only over arithmetical formulae φ, whose non-logical symbols are just 0, S, '+', '·', and lower case letters as variables ranging over the natural numbers. The new axiom schema represents a countably infinite set of axioms.

 

The axioms of PA are:

 

 

for any formula in the language of PA.

 

 

PA does not require the predicate "is a natural number" because the universe of discourse of PA is just the natural numbers N. While no explicit existential quantifiers appear in the above axioms, four tacit quantifiers of that nature follow from the closure of the natural numbers under zero, successor, addition, and multiplication; and there may be implicit existential quantifiers in the axioms of induction within the

 

Existence and uniqueness

A standard construction in set theory shows the existence of a Dedekind-Peano structure. First, we define the successor function; for any set a, the successor of a is S(a) := a ∪ {a}. A set A is an inductive set if it is closed under the successor function, i.e. whenever a is in A, S(a) is also in A. We now define:

 

0 := {}

N := the intersection of all inductive sets containing 0

S := the successor function restricted to N

The set N is the set of natural numbers; it is sometimes denoted by the Greek letter ω, especially in the context of studying ordinal numbers.

 

The axiom of infinity guarantees the existence of an inductive set, so the set N is well-defined. The natural number system (N, 0, S) can be shown to satisfy the Peano axioms. Each natural number is then equal to the set of natural numbers less than it, so that

 

0 := {}

1 := S(0) = {0}

2 := S(1) = {0,1} = {0, {0}}

3 := S(2) = {0,1,2} = {0, {0}, {0, {0}}}

and so on. This construction is due to John von Neumann.

 

This is not the only possible construction of a Dedekind-Peano structure. For instance, if we assume the construction of the set N = {0, 1, 2,...} and successor function S above, we could also define X := {5, 6, 7,...}, x := 5, and f := successor function restricted to X. Then this is also a Dedekind-Peano structure.

 

The lambda calculus gives another construction of the natural numbers that satisfies the Peano axioms.

 

Two Dedekind-Peano structures (X, x, f) and (Y, y, g) are said to be isomorphic if there is a bijection φ : X → Y such that φ(x) = y and φf = gφ. It can be shown that any two Dedekind-Peano structures are isomorphic; in this sense, there is a "unique" Dedekind-Peano structure satisfying the Peano axioms. (See the categorical discussion below.)

 

Categorical interpretation

The Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1 is the following category:

 

The objects of US1 are all ordered triples (X, x, f), where X is a set, x is an element of X, and f is a set map from X to itself.

For each (X, x, f), (Y, y, g) in US1, HomUS1((X, x, f), (Y, y, g)) = {set maps φ : X → Y | φ(x) = y and φf = gφ}

Composition of morphisms is the usual composition of set mappings.

The natural number system (N, 0, S) constructed above is an object in this category; it is called the natural unary system. It can be shown that the natural unary system is an initial object in US1, and so it is unique up to a unique isomorphism. This means that for any other object (X, x, f) in US1, there is a unique morphism φ : (N, 0, S) → (X, x, f). That is, that for any set X, any element x of X, and any set map f from X to itself, there is a unique set map φ : N → X such that φ(0) = x and φ(a + 1) = f(φ(a)) for all a in N. This is precisely the definition of simple recursion.

 

This concept can be generalised to arbitrary categories. Let C be a category with (unique) terminal object 1, and let US1© be the category of pointed unary systems in C; i.e. US1© is the following category:

 

The objects of US1© are all ordered triples (X, x, f), where X is an object of C, and x : 1 → X and f : X → X are morphisms in C.

For each (X, x, f), (Y, y, g) in US1©, HomUS1©((X, x, f), (Y, y, g)) = {φ : φ is in HomC(X, Y), φx = y, and φf = gφ}

Composition of morphisms is the composition of morphisms in C.

Then C is said to satisfy the Dedekind-Peano axiom if there exists an initial object in US1©. This initial object is called a natural number object in C. The simple recursion theorem is simply an expression of the fact that the natural number system (N, 0, S) is a natural number object in the category Set.

 

Metamathematical discussion

These axioms are given here in a second-order predicate calculus form. See first-order predicate calculus for a way to rephrase these axioms to be first-order.

 

Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers.

 

On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic.

 

If one replaces the last axiom with the schema:

 

If P(0) is true; and for all x, P(x) implies P(x + 1); then P(x) is true for all x.

for each first order property P(x) (an infinite number of axioms) then although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrarily large cardinality - by the compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization; by an "upward Löwenheim-Skolem theorem", there exist models of all cardinalities. Moreover, if Dedekind's proof that the second order Peano Axioms have a unique model is viewed as a proof in a first order axiomatization of set theory such as Zermelo–Fraenkel set theory, the proof only shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism; nonstandard models of set theory may contain nonstandard models of the second order Peano Axioms.

 

When the axioms were first proposed, people such as Bertrand Russell agreed these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent. If a proof can exist that starts from just these axioms, and derives a contradiction such as P AND (NOT P), then the axioms are inconsistent, and don't really define anything. David Hilbert posed a problem of proving consistency using only finite logic as one of the problems on his famous list.

 

But in 1931, Kurt Gödel in his celebrated second incompleteness theorem showed such a proof cannot be given in any subsystem of Peano arithmetic. Furthermore, we can never prove that any axiom system is consistent within the system itself, if it is at least as strong as Peano's axioms, instead one must prove the consistency of the system using a different axiomatic system. Although it is widely believed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this is not clear, and depends on exactly what one means by a finitistic proof: Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. As Gentzen has explained it himself: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers). Whether or not this really counts as the "finitistic proof" that Hilbert wanted is unclear: the main problem with deciding this question is that there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

 

Most mathematicians assume that Peano arithmetic is consistent, although this relies on either intuition or on accepting Gentzen's proof. However, early forms of naïve set theory also intuitively looked consistent, before the inconsistencies were discovered.

 

Founding a mathematical system upon axioms, such as the Peano axioms for natural numbers or axiomatic set theory or Euclidean geometry is sometimes labeled the axiomatic method or axiomatics.

 

 

Can we please please try and keep the questions at a level that might be found in a Wednesday night pub quiz.... :blink:

 

 

In the meantime....

 

Which German line lay opposite the French Maginot Line?

 

Pub Quiz!!!

 

C'mon peeps :P:):):):):):blink::huh::)

 

The Golems next question nearly got lost (and we appear to have the same quiz night! :DB):D:P )

 

"In the meantime....

 

Which German line lay opposite the French Maginot Line?"

Edited by Nediam
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Simply Paul

+Simply Paul

Posted
Posted

Both, actually.

 

To quote from a well-regarded online source I just checked, "However, in English, Siegfried line more commonly refers to the similar World War II defensive line, built during the 1930s, opposite the French Maginot Line, which served a corresponding purpose. The Germans themselves called this the Westwall, but the Allies renamed it after the First World War line. This article deals with this second Siegfried line."

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The Bongtwashes

+The Bongtwashes

Posted
Posted

Both, actually.

 

To quote from a well-regarded online source I just checked, "However, in English, Siegfried line more commonly refers to the similar World War II defensive line, built during the 1930s, opposite the French Maginot Line, which served a corresponding purpose. The Germans themselves called this the Westwall, but the Allies renamed it after the First World War line. This article deals with this second Siegfried line."

Thanks SP for a slightly more concise explanation than we had for Peano's Postulates (wasn't he in a Carry On movie?). :laughing:

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Simply Paul

+Simply Paul

Posted
Posted

I'm feeling confident enough to Auto-Ding myself. If The Golem aducitates otherwise, I'm happy to give up my claim to it.

 

With washed-up goods in the news today, and with my eye still firmly on the Western Isles, you'll already know the fictional island of Todday features in the classic 1949 film of coastal scavenging, Whisky Galore! It's based on the true story of the SS Politician (SS Cabinet Minister in the film), which sunk close to Eriskay in the Outer Hebrides in February 1941

 

So, the question: Which other film does Todday feature in?

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The Forester

+The Forester

Posted
Posted

In Nigeria I was obliged to use a decrepit vessel called "The Maggie". Unlike landlubbers, mariners don't call ships "The" so I was surpised that she wasn't simply named "Maggie". When I asked it was explained to me that she, like her sistership "African Queen", was named after a film. I seem to recall that the film was some kind of sequel or prequel to Whisky Galore.

 

My guess is that the second Todday/Barra film was 'The Maggie'

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Simply Paul

+Simply Paul

Posted
Posted

Ding! Rockets Galore it is. Regarded as a rather weak follow-up in the UK on its release in 1957, it was well received in the US under the title 'Mad Little Island' (Whisky Galore! was 'Tight Little Island' over there), possibly because Barra looks beautiful in Eastmancolor (WG was filmed in black and white). It features a young Ronnie Corbett and reunites much of the cast of Whisky Galore! under a different director, Michael Relph. The siting of a rocket range test launch centre on Benbecula in the 50's, as The Forester says, must have influenced the film. So now you know :rolleyes:

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The Forester

+The Forester

Posted
Posted
(Whisky Galore! was 'Tight Little Island' over there)

 

That dispels an urban legend which I'd always believed. I'd heard that the film was called 'Whisky a go-go' in the US and that someone opened a dance bar under that name as a result. The bar's dancing girls were called "go-go girls" and so "go-go dancing" was launched.

 

Oh well, I'll have to think of a Q now.

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Simply Paul

+Simply Paul

Posted
Posted (edited)

The two at the end of Ursa Major (aka the Big Dipper or Plough) are Castor and Pollux, I think. If you draw a line through them, and continue it up, Polaris is the first bright star fairly close to the line which you come to.

 

Edited for my shocking spelling. Anyway, those aren't the right names of the stars, I now see. As I've checked a book for this, I won't now jump in with the right names!

Edited by Simply Paul
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