Acknowledgements:

I would like to acknowledge the contributions of others to this work.

First Twiffy, another mathematician and an occasional poster here at ILP, referred me to the book “INTRODUCTION TO SET THEORY” by Karel Hrbacek and Thomas Jech. I was unfamiliar with the foundational mathematics of ZFC and he kindly helped me through with answers to some of my questions. Additionally, in the body of this work I intend to include some introductory material form “Abstract and Concrete Categories: The Joy of Cats” by Jiri Adamek, Horst Herrlich, and George E. Strecker.

I would also like to thank MusicDreamer for his work editing this introduction.

Preface:

This section is a purely personal and subjective impression; and the reader not interested in fluff can skip ahead to the section entitled Background.

In my youth and naiveté, I considered Bertrand Russell one of the greatest minds of all time, perhaps even a living God. Twiffy echoing similar if somewhat toned down sentiments, called Russell “The Man”.

The problem was that in my zeal and admiration for the man and the quest for truth that he represented, I stood ready to abandon my beliefs in God and Christianity. I opened his book entitled “Why I am not a Christian”, awaiting the shear blissful revelation of overwhelming logic, based on unassailable principals, to tear me away from my antiquated and foolish two thousand year old system of beliefs.

You can read Russell’s book and come to your own conclusions. But for me, after many instances of screaming out loud, Russell was no longer a God.

I also read Russell’s book (or much of it) “The History of Western Philosophy”. Even though it was obviously just a survey book, I remember thinking that some of his comments were inconsistent with his description of the base material. More disappointment.

Many years later, while investigating the foundations of mathematics, I encountered Wittgenstein and read about the relationship between the two men. The net result was that Russell appeared weak and willing to simply kowtow to the loathsome Wittgenstein. (Despite the fact that, in my opinion, Russell’s instincts were closer to the truth in many cases).

Again I was disappointed with Russell.

Now in preparation for this analysis, I have read much of the Principia Mathematica source material, and I am blown away by its brilliance. Despite some minor flaws, this is Russell in his full glory and Principia Mathematica is an incredible masterpiece.

Personally, I am more impressed with Russell’s intellectual ingenuity and rigor in this work than with Gödel’s very impressive work “On Formally Undecidable Propositions Of Principia Mathematica And Related Systems”.

Russell expressed the fact that he was intellectually exhausted after his work on this book, and that he never recovered. I don’t think that anyone, after reading that work, would question the fact that constructing “Principia Mathematica” would be an intellectually exhausting experience.

I feel that I now understand why Russell never lived up to my early expectations.

Background:

Leopold Kronecker:

I will begin with Leopold Kronecker (Wikipedia starts with Frege*). Kronecker was born December 7, 1823 and died December 29, 1891. He believed that the Counting numbers were a gift from God and his most famous quote “God created the Integers. All the rest is the work of man.” was in part taken by Stephen Hawking as the title of his 2005 edited collection of the important works in mathematics.**

The important thing here is that Kronecker was very conservative and would not acknowledge certain types of infinite extensions. These extensions would be numbers that were only defined by sequences of Rational numbers that appear to converge on a given number. Such an extension is defined by the sequence (1 + 1/n) raised to the nth power. The first number of this sequence is 2, the second number is 1-1/2 squared or 2- 1/4, the third number is (4/3) raised to the third power or 4 x 4 x 4 / 3 x 3 x 3 or 64 / 27, approximately 2.3704, et cetera. The reader might notice that each term in this sequence is a Rational number (it can be expressed as a whole number plus a fraction).

The difference between the successive terms in this sequence get arbitrarily small as n gets large. This type of sequence is called a Cauchy sequence. The problem is that there is no guaranty that the limit of this sequence, which is generally designated by the letter e, actually exists as a Real number (if at all). As an example of a sequence of Rational numbers which gets arbitrarily small and yet the limit does not exist, consider the set (0, 1) which is the set of Real numbers from 0 to 1 exclusive of the numbers 0 and 1. Then the sequence 1 / (n + 1) will get arbitrarily close to 0, and yet the limit of 0 does not exist in the set (0, 1).

Unlike Pythagoras, Kronecker did allow for the existence of numbers that could not be expressed as the ratio of two Integers, namely those numbers that could be constructed as the roots of polynomial equations; but he did not accept the existence of a transcendental number. As a practical matter he did not believe in numbers such as e (a super cool number sometimes called Euler’s number) and pi, the ratio of the circumference to the diameter of a circle.

For a pure simple expression of the joy that the number, e, brings to me and many others, see my homage at:

viewtopic.php?f=4&t=144168&hilit=+Homage

His constrained view of mathematics is somewhat consistent with both L. E. I. Brouwer’s development of Intuitionism and Wittgenstien’s mathematical foundations***.

Kronecker was the foil to two of the most important men in mathematics. One was Karl Weierstrass, who formulated modern calculus, and freed it from the illogical concept of the infinitesimal which was imbedded in the calculus of both Newton and Leibnitz. The other was Georg Cantor the father of Set Theory. The relationship between Kronecker and these two was often bitter and very emotionally charged.

Georg Cantor:

Georg Cantor was born circa March 3, 1845 and died January 6, 1918. He was a student under his great nemesis Leopold Kronecker and a friend at that time of Karl Weierstrass when they were at the University of Berlin. Cantor also spent a year at Gottingen which by that time had become the Mecca for mathematicians.

In 1891 Cantor published his paper on the diagonal argument, which he used to prove that uncountable sets, sets that are more numerous than the Counting numbers, exist. (This was not his first proof of the matter, but it is very elegant and it can be used to prove other important theorems).

There is more on the diagonal argument at:

http://en.wikipedia.org/wiki/Cantor%27s ... l_argument

One of the most important theorems, proven using this tool, was that the cardinality (a measurement of the number of elements in a set) of the set of all subsets of a given set, A, was strictly larger than the cardinality of the set A itself. It is a nearly immediate consequence of this fact that:

The set of all sets cannot exist.

The above statement is extremely important in the development of Principia Mathematica.

Cantor’s Fun Facts:

The cardinality of the Reals is greater than the cardinality of the Natural numbers. (One infinity can be more numerous than another infinity).

The cardinality of the Transcendental numbers, Real numbers that are neither Integers nor roots of polynomial equations, is greater than the cardinality of all the other Real numbers combined. The only transcendental numbers that I can name are e and pi, and yet they are far more numerous than any other commonly constructed subset of the Reals (such as the Rational numbers, numbers that can be expressed as p/q where p and q are whole numbers).

The Cantor set, which is constructed by dropping successive middle thirds from the connected residual sets starting with the unit interval [0, 1], see http://en.wikipedia.org/wiki/Cantor_set for a nice pictorial representation of the construction, results in what is both technically and intuitively called dust. The fun fact is that the dust is more numerous than all the fractions from 0 to 1.

The scholarly reaction at the time can largely be summarized as follows:

“Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".[”

The above quote is from Wikipedia located at:

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

My favorite quote in this section of Wikipedia is from David Hilbert, the person that probably had the greatest impact on the course of mathematics in the twentieth century, and it reads as follows:

"No one shall expel us from the Paradise that Cantor has created."

The bulk of Cantor’s theorems are taught as part of almost all modern advanced mathematics today.

How do we deal with infinity?

From my observation, many people are simply overpowered by the concept of infinity. Some associate it with the Divine. Some claim it to be a logical uncertainty or illogical.

For me, like the vast majority of mathematicians, I stay clear of these considerations and work with what seems like a reasonable tool.

Now for that tool.

First we establish a conventional yardstick or unit of measurement. Here I will use the Counting numbers (The whole numbers starting at 1).

Then we observe that we can count the number of elements in a finite set by associating the elements in that set with subsets of the Counting numbers.

For example:

Consider the set {A, B, C}. We can associate {A, B, C} with the set {1, 2, 3}, where A is associated with 1 and B is associated with 2 and C is associated with 3. Since we are starting at 1, we know that this association will yield 3 as the highest number.

Similarly, the set {10, 15, 20, 45} can be associated with the set {1, 2, 3, 4}. Again the first element 10 is associated with the first element of the counting numbers 1 and the second element of the set, 15, is associated with the second element of the Counting numbers, 2, the third element of the set, 20, is associated to the third Counting number 3, and finally the fourth element of the set, 45, is associated to the fourth Counting number, 4.

In general if each element of a set can be associated to exactly one ascending element of the Counting numbers, then we can say that the highest Counting number is the number of elements in that set.

Now we abstract the idea that we can find the number of elements in a finite set, by finding this same type of association and applying it to infinite sets using all of the Counting numbers as our yardstick.

We simply define the number of elements of an infinite set S to be countable if and only if there is a function f such that all of the elements of the Counting numbers are uniquely mapped to all of the elements of S.

As an example let’s examine the set of Natural numbers, or the Counting numbers plus the element 0. Here we let the association be the function f mapping the Counting numbers, C, to the Natural numbers, N, defined as f(n) = n - 1. This function maps unique numbers in C to unique numbers in N and every element of N is such that if n is an element of N then n = f(m) for some m. The reader can prove the last assertion for fun.

This means that despite the fact that N has one more element than C, it has the same cardinality as C.

On the other hand smaller sets such as the odds can have the same cardinality as C. The function f(n) = 2n - 1 gives an appropriate association. Similarly the function f(n) = 2n is an appropriate function for mapping C to the evens. An inventive reader might be able to show that the ordered pairs (m, n) where n and m are Integers (whole numbers including the negative numbers) have the same Cardinality as the Counting numbers.

This means that “smaller” sets and “larger” sets can all have the same Cardinality as the Counting numbers.

All sets that have the same Cardinality as the Counting numbers are said to be countable.

As mentioned above Cantor showed that the Cardinality of the Real numbers is greater than the Cardinality of the Counting numbers. The Reals are said to be uncountable.

The Foundation Makers:

From about 1860 through 1930 many of the sharpest minds worked on establishing axiomatic foundations for arithmetic, set theory and geometry.

A simple, and likely incomplete, list in chronological order would be:

1) Hermann Grassmann in 1861 gave the first axiomatic foundation for Arithmetic.

2) Charles Sanders Pierce in 1881 gave an axiomatic foundation of the Natural numbers.

3) Gottleg Frege in 1884 gave a preliminary foundation for arithmetic and in 1893 a formal axiomatic foundation. This foundation relied heavily on naïve set theory.

4) David Hilbert in 1899 produced an axiomatic foundation of Euclidean Geometry.

5) Richard Dedekind 1897 and Giuseppe Peano, relying heavily on Dedekind’s work, in 1898 gave an axiomatic foundation of the Counting numbers. This foundation is now referred to as the Peano axioms.

6) Ernst Zermelo in 1908 gave an axiomatic foundation for set theory. It was later expanded by Abraham Fraenkel and Thoralf Skolem.

7) LEJ Brouwer in 1908 informally lays out the foundations of Intuitionism in his paper entitled “The untrustworthiness of the principles of logic”.

8) Bertrand Russell and Alfred Whitehead in 1910 copyrighted Principia Mathematica.

9) John Von Neumann in 1925 and in an extension in 1928 provided an alternative axiomatic foundation to set theory. This was later expanded by Paul Bernays and Kurt Gödel.

10) Alfred Tarski in 1926 published an axiomatic foundation of Euclidean Geometry.

A number of the authors above affected others. Grassmann affected Dedekind and Peano. Pierce, Frege, and Peano all affected Russell. Von Neumann, well aware of first order logic, was affected by Zermelo et al of ZFC fame. And Tarski, aware of the concept of completeness, was affected by Hilbert and Gödel.

The question to ask is: Why do we need axiomatic systems in the first place?

Almost all of us learn about numbers by being shown examples of two apples, two oranges, and two bananas (or whatever examples are handy). Then we abstract the concept of twoness for the number 2. It seems simple.

The problem, with this system, starts to show itself when the numbers start to become large.

How do we know that by adding apples to the bucket, we will always get more apples? After a while, in real life, the apples will rot and ferment.

How do we know that there is not a largest number? After all about this time Einstein showed that, in a vacuum c was a largest speed. We also know that someday, if we keep adding apples to a bucket, we will run out of apples.

Basically we need to take a leap of faith.

We need to assume that if n is a number then n + 1 is a number. We also need to assume that n + 1 will never be a number that is less than or equal to n.

The Foundations:

Before proceeding I would like to mention that each or the following proposed axiomatic foundations are taken from referenced sources. However, the blue comments are personal observations.

I would also like the reader to understand the frame of mind that, from my perspective, is most useful in analyzing various foundations. First we need to remember that we are assuming that there are not yet such things as numbers. This implies that there is not yet such things as addition or multiplication. After the axioms are presented, we need them to match our current, abstracted, understanding of how numbers should work.

Peano Axioms:

These are not historically accurate but they are logically equivalent and they are found in Wikipedia. Additionally, it is expedient to use. I have deleted the Wiki commentary as I believe it to be inappropriate for this audience.

1. 0 is a natural number.

The next four axioms are logical identities.

2. For every natural number x, x = x.

3. For all natural numbers x and y, if x = y, then y = x.

4. For all natural numbers x, y and z, if x = y and y = z, then x = z.

5. For all a and b, if a is a natural number and a = b, then b is also a natural number.

The remaining axioms are specifically about numbers.

6. For every natural number n, there exists a successor function S such that S(n) is a natural number. Remark: as numbers get larger they must remain numbers.

7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0. Remark: numbers cannot reach a highest number and then “circle back”.

8. For all natural numbers m and n, if S(m) = S(n), then m = n. Remark: The successor function S must assign different numbers to different successor values.

9. If K is a set such that:

1. 0 is in K, and

2. for every natural number n, if n is in K, then S(n) is in K,

then K contains every natural number. Remark: If 0 is in a larger set, and all the successor numbers, S(n), are in K then all the natural numbers are in K. This implies that there are no natural numbers that are not represented by S(n) for some n because we can set K to be all the natural numbers.

The Natural numbers derived from the Peano axioms are (assumed to be):

0, S(0) or 1, S(1) or 2, S(2) or 3, …, m, S(m) or m +1, …

Notes:

Despite the fact that the Peano axioms were, if I recall correctly, included in my grade school text book they have significant problems. One of which is that if S(0) = 1, and S(1) = 2 et cetera, and there is no guarantee of that, we would have the problem that the function S would have as a domain the natural numbers. This means that we would be defining the natural numbers by means of the natural numbers.

Additionally, since we don’t know what S(n) is, I invite the reader to consider the possibility that S(n) might be 2n.

Set Theory Axioms:

ZFC or Zermelo-Fraenkel axioms with the axiom of Choice. It is important to remember here that every element of any set is defined to be a set. i.e. a set cannot consist of things that are not sets. For example we might commonly think that a collection such as {“a”, “5”, “book”} would be a set. However, ZFC does not consider this collection to be a set “a” is not a set, the number “5”, at least as defined by our common abstracted definition or the Peano axioms is not a set, and the word “book” is not a set.

These axioms are from the book “INTRODUCTION TO SET THEORY” by Karel Hrbacek and Thomas Jech. If the reader wishes to follow on Wikipedia the axioms are listed at:

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

The reader should be aware that there are differences in order, names and even number of axioms.

They are:

1. The Axiom of Existence. There is a set with no elements. Remark: This is analogous to 0.

2. The Axiom of Extensionality. If every element of X is an element of Y and every element of Y is an element of X, then X = Y. Remark: Two sets that have the same elements are equal.

3. The Axiom Schema of Comprehension. Let P(x) be a property of x. For any set A, there is a set B such that x is an element of B if and only if x is an element of A and P(x). Remark: This axiom states that for a Set A there exists a subset B where x is an element of B if and only if P(x) is true. The reader might wonder why this is an axiom schema. The reason is because it represents many axioms dependent on the specific property P. Probably the most important reason for this axiom is that it avoids the Russell Paradox.

4. The Axiom of Pair. For any A and B there exists a set C such that x is an element of C if and only if x is an element of A or x is an element of B. Remark: This implies that if A and B are sets, then the set {A, B} exists and it is unique.

5. The Axiom of Union. For any set S, there exists U such that x is an element of U if and only if x is an element of A for some A contained in S. Remark: This is a somewhat odd way of saying if M and N are sets then x is an element of the set consisting of the union of M and N if and only if x is an element of M or x is an element of N.

6. The Axiom of Power Set. For any set S, there exists a set P such that x is an element of P if and only if x is contained in S. Remark: P consists exclusively of all of the subsets of S.

7. The Axiom of Infinity. An inductive set exists. Remark: An inductive set A is defined to contain the empty set and at least one other x. In addition the sets I(n) = {{I(n-1)} union x} is contained in A where I(0) is defined as x. Here I(1) = {{x} union x}. If one does not accept the logical law of the excluded middle, such as the constructivists Brouwer and Wittgenstein, then the axiom of Infinity has nothing to do with infinity.

8. The axiom of Choice. For every set X there exists an ordered set I and a function f such that if x is an element of X, then x = f(i) for some i in I. Remark: If x is an element of X, then we can use x in existence proofs, (because we can find x).

The Natural numbers derived from the set theory ZFC are:

The empty set or 0, {0} or 1, {0, {0}} or 2, {0, {0} ,{0,{0}} or 3 …, n or {0, 1, 2, …, n-1}, …

Notes:

The axiom of existence gives us no perspective or background information on what a set of no elements might be and thus, for some people including myself, this makes the axiom of existence suspect. The Wikipedia authors avoid using this axiom and always define “a set of no elements” with respect to a background or Universe.

As is pointed out in the book “Introduction To Set Theory” by Hrbacek and Jech some of the axioms are redundant. The authors write “For example, the Axiom of Existence and the Axiom of Pair can be proved from the rest”. (Personally I do not agree with this assessment because the set with no elements for the other axioms will require a universe from which to work. The axiom of existence has no such restriction. I prefer that the set with no elements be defined by a universe; but that does not mean that the Axiom of existence can be derived from the other axioms).

Hrbacek and Jech also observe “... the Axiom of Choice asserts that certain sets (the choice functions) exist without describing those sets as collections of objects having a particular property. Because of this, and because of some of its counter intuitive consequences… some mathematicians raised objections to its use”.

The Axiom Schema of Comprehension and the Russell Paradox:

Because the set of all sets cannot exist, anyone trying to deal with an axiomatic approach must try to construct sets that do not have themselves as members.

Zermelo noticed this and constructed the Russell Paradox prior to Russell. But he was working a Gottingen at the time and simply communicated it to the mathematicians that were there.

Frege, working independently tried to exclude the set of all sets by limiting sets to sets that do not contain themselves. Russell, who had previously noticed that the phrase “… sets consisting of sets that that do not contain themselves.” was paradoxical wrote about it to Frege as he was about to go to press with his now debunked foundation. (It should be noted that Russell discovered the paradox by trying to prove that Cantor’s proof, that there was no highest Cardinality, was flawed. Frege’s contribution could be construed as an afterthought).

Russell’s paradox attacks the assertion that there exists a set R defined to be the set of all sets that are not members of themselves. If the assertion were true then there must be some set y, such that for every x, x is an element of y if and only if x is not an element of x.

To see that this is a logical fallacy we substitute y for x to get:

y is an element of y if and only if y is not an element of y. This is a contradiction.

By contrast if we use the Axiom Schema of Comprehension we get:

Let P(x) be the property x is not an element of x then the axiom schema of comprehension says: For any set A, there is a set B such that x is an element of B if and only if x is an element of A and x is not an element of x.

Here there is an additional set A and x is required to be in A as well as be in B. Now if we look at B and substitute it for x we get B is an element of A and B is not an element of B. There are many examples of sets that fit this criteria.

The axiom of Comprehension places a restriction on set membership which intuitively seems unnecessary, though it does provide an escape from Russell’s Paradox.

It should be mentioned that while Wikipedia writes: “ZFC has remained the canonical axiomatic set theory down to the present day”, it is my experience that the overwhelming majority of mathematicians define sets much more broadly. More about this in the body of the text.

NBG Set Theory:

This rather obtuse name is an acronym for von Neumann, Bernays and Gödel.

In his book “Surely Your Joking, Mr. Feynman” Richard Feynman refers to John von Neumann as “the great mathematician”. This probably should be enough to give you an idea of his stature. However I will mention that he is considered the father of modern game theory and had a permanent position at the Institute for Advanced Study in Princeton. Wikipedia states: Von Neumann had a very strong eidetic memory. If you have been reading the Wikipedia biographies of the people that created the foundations, you definitely should read this one.

Paul Bernays is the only other name that you might not know. His PhD examiner was Ernst Zermelo and in 1917 he worked with David Hilbert on the foundations of Arithmetic. His major contribution was to change von Neumann’s Foundation from function based to set based.

I assume that everyone knows Gödel.

Since, Wikipedia claims that theorems in ZFC are valid if and only if they are also true in NGB I will not go into an analysis of the axioms. However, it should be mentioned that NBG Set Theory has the advantage of being stated finitely. e.g. the axiom of Schema of Comprehension actually states an infinite number of axioms, depending on P, whereas NBG avoids this problem.

Intuitionism:

The main advocate of Intuitionism is L.E.I. Brouwer.

Brouwer:

“This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness.”

The above quote is from:

http://www.ams.org/journals/bull/2000-3 ... 0802-2.pdf

The reading of this document is interesting from the point of view of a critique of Kantian philosophy.

Since one can only start from a first moment, I believe that Brouwer’s concept of numbers, much like our historical concept, starts not with 0 but with 1. The Peano numbers also historically started with 1. Additionally, much like the Peano numbers, Brouwer’s Intuitionism is clearly ordered and numbers may be considered as Ordinals. i.e. they really represent a first number, a second number and so on.

The Counting numbers 1 or the cognitive recognition of the passage of time, 2 or the recognition of an additional passage of time, 3 the additional passage of time from the second, …., .

Notes:

Much like the Peano numbers, Brouwer’s concept of numbers is clearly ordered and numbers may be considered as Ordinals. i.e. they really represent a first number, a second number and so on.

A major drawback to the Intuitionist construction is that they do not wish to allow for an actual infinity in their mathematical constructions which ultimately led to a rejection on the law of logic called the excluded middle. This consequence of this rejection is that proofs by contradiction cannot be allowed. Such a restriction would void a major part of mathematics.

It is also important to note that the Intuitionists, like Kronecker, do not accept the existence of e or pi.

Gödel showed that if the principle of the excluded middle was invalid then the logic of any system based on this exclusion was infinite valued. Effectively, Intuitionists cannot avoid infinity.

Reference:

http://en.wikipedia.org/wiki/Many-valued_logic

It should be noted that even though the Intuitionists have failed at their effort to keep infinity out of mathematics, it is my observation that constructive proofs (proofs that do not rely on the construct of assuming a negation and then proving that negation to be false) are preferred by Wikipedia and many other mathematicians.

The final two axiomatic systems that I will simply mention are Hilbert’s axiomatization of geometry, and Tarski’s axiomatization of geometry.

Hilbert’s axiomatization of Geometry included 21 axioms, but it was later discovered that one axiom was redundant and the number of axioms was ultimately reduced to 20 axioms.

Alfred Tarski also axiomatized Geometry. His axiomatization included 10 axioms and one axiom schema.

These two axiomatizations of Geometry are significantly impacted by the ramifications of Principia Mathematica.

* It should be noted that there is a massive difference of opinion between the Wikipedia authors and I. Grattan – Guinness a highly respected historian of mathematics. In his book “The Search for Mathematical Roots 1870 – 1940” he talks about today’s fictional character of Frege and deems him Frege’. Wikipedia writes, “One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, …”

Located at http://en.wikipedia.org/wiki/Principia_Mathematica

Grattan – Guinness would almost certainly dispute this statement.

In order to stay clear of the conflicting points of view I will generally refer to Frege only as he can be evidenced through Principia Mathematica. However, in almost all other cases, the two accounts of the main characters involved with the mathematical foundations are similar enough that, in my opinion, the reader can reliably refer to the Wikipedia accounts.

** It is interesting to me that Hawking, who had met with a pope on the subject of the Big Bang, rejected the idea that somehow the biblical account of creation could be reconciled with the theory of the Big Bang. Yet he chooses this very odd title for his book.

*** It might be noted that Wittgenstein, considered by many to be the greatest philosopher of the 20th century, with a reputed IQ of 190, considered his work on the foundation of mathematics to be his greatest contribution to philosophy. Personally I am skeptical on all counts. I have seen his work on this matter and it appears simplistic and flawed on the face of it. Certainly (with the exception of Principia Mathematica), he is never mentioned in any of the principal works on the foundations of mathematics that I have ever seen.