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### SI Prefixes

The original SI (systeme international, or "Metric" system) in 1793
had only the prefixes kilo, hecto, deca, deci, centi, and milli. In
1795 myria was added but later became deprecated. mega dates from
the late 1800's and was officially adopted in France in 1919. During
the 1900's kilomega and megamega were used but it was eventually
decided these needed their own prefixes^{3}; giga, nano and
tera, pico were adopted in 1960, femto and atto in 1964,
peta and exa in 1975, and zetta, zepto and yotta, yocto in
1991. These extensions have been added mainly for the convenience of
scientists. For example, in 1993 some researchers had to refer to
units of 10^{-21} volts but they didn't yet know about the prefix
zepto so they called it "milliattovolt". There have always been
fields where very small or very large values are expressed simply as
big exponents of 10, or where a field-specific and somewhat arbitrary
unit such as the parsec or electron-volt is used. The standard unit of
atomic mass (1/12 the mass of a Carbon atom, or roughly the mass of a
proton or a neutron) is 1.66×10^{-27} kg or 1.66 yg (1.66
yoctograms). Quasars have been discovered that are about 125
yottameters away. When volumes or weights are involved the units are
even more often found to be insufficient — the Earth weighs about
6000 Yg (6000 yottagrams), and the mass of an electron is about
0.00091 yg (yoctograms). By comparison, the diameter of the Earth and
of an atom (both one-dimensional measurements) are both easily handled
with the older kilo- and pico- prefixes.

The following "SI prefixes" are deprecated or obsolete: myria, kilomega, and megamega.

The official SI prefix names "peta" and "exa" are probably derived
from "pente" and "hexa" (five and six in Greek)^{3}, however, the
official BIPM (Bureau International des Poids at Mesures)
website does not give an explanation.

"Zetta" and "yotta" are derived from "septo" and "octo", which are
quasi- number names. Quoting BIPM:^{13}

The names zepto and zetta are derived from septo suggesting
the number seven (the seventh power of 10^{3}) and the letter "z" is
substituted for the letter "s" to avoid the duplicate use of the
letter "s" as a symbol. The names yocto and yotta are derived from
octo, suggesting the number eight (the eighth power of 10^{3}); the
letter "y" is added to avoid the use of the letter "o" as a symbol
because it may be confused with the number zero.

If BIPM decides to adopt further prefixes for 10^{27} and 10^{30} and
their reciprocals 10^{-27} and 10^{-30}, they will probably adopt
something vaguely resembling names for nine and ten for a similar
reason — perhaps something like novetta, novemo, decetta and
decemo. If so they would almost certainly be assigned two-letter
abbreviations such as "No-", "De-", "no-" and "de-" because
N-, n- and d- are already used for other prefixes.

### Joke, Hoax, and Sincere (but Ad-Hoc) SI Prefix Proposals

The following prefixes have all been exposed as jokes or
hoaxes: bronto, cuppa, dea, ento, fito, harpo, hepa,
lotta, lotto, nea, otta, revo, syto, tredo, una,
xera, zeppo, and zuppa^{3},^{27},^{28},^{29},^{33},^{34}.
Several of these were perpetuated by Internet rumors (compare two
posts by Alex Lopez-Ortiz ^{4},^{11}), and many go back before
widespread use of the Internet^{12}.

One of the earliest examples was apparently a reaction to the
silly-sounding and real prefixes yocto and zepto. In 1993, as a
joke that was reportedly well received on USENET, Morgan Burke^{27}
proposed harpo for 10^{-27} and groucho for 10^{-30} (and
therefore harpi for 10^{27} and grouchi for 10^{30})^{29}.

In addition, the following are simply bogus ad-hoc personal ideas:
luma, lunto, mikto, minga, nekto, nena, ocha, otro,
pekro, pepta, quekto, quexa, rimto, rinta, sorta,
sotro, treda, trekto, uda, udeka, udeko, unto,
vendeka, vendeko, vunda, vunkto, weka, weko, wekta,
wekto, xenno, xenta, xona, and xonto^{10}.

Oxford professor Jeff K. Aronson has suggested extending beyond zetta, zepto and yotta, yocto with xenta, xenno, wekta, weko, vendeka, vendeko, udeka, udeko based on the idea that the 'Z' and 'Y' prefixes would continue backwards through the English or modern Latin alphabet.

An essentially equivalent proposal was made by Paul Shuch in a letter
to Science News ^{35}, suggesting acto=10^{-96}, atta=10^{96},
etc. through wocto=10^{-30}, wotta=10^{30}, xocto=10^{-27}, and
xotta=10^{27}.

Using a similar idea, Jim Blowers^{10} says:

The pattern here is that we go backwards from the beginning
of the alphabet [shouldn't this be "end of the alphabet"? -ed],
starting with z and y, and we follow it up with an
alteration of the Greek or Latin for the next number. According to
this pattern, the next ending ["prefix" -ed] should be xona-,
since x comes before y in the alphabet, and 9 is noni- in Latin.
Similarly, 10^{30} should be weka-, since w precedes x and 10 is
deka in Greek.

He goes on to list a large number of prefixes starting with Xona-,
Weka-, Vunda-, Uda-, Treda-, Sorta-, ... One-letter
abbreviations are used if unambiguous, otherwise another letter is
added, e.g. TD- for the Treda- prefix. He goes as far as 10^{63}
using an L- prefix, based on the 26-letter "Latin" alphabet used in
several European languages including English, although the original
classical Latin language had no U or W.

Any of these might be (intentionally of mistakenly) used by others who are sincere, but unaware that the prefix they are using is unofficial, ad-hoc or worse. For example, one reader wrote to me informing me of the usage of Xera- by a Dr. Laurent Alexandre.

### Sources

Here is a collection of URLs that are related to the above topics, but which I did not actually use as sources. Most of these are from the beginning of my work on this page (around 1999 or 2000). These pages concern names of large numbers, SI units, and similar things.

(Sources:

Borislav Manolov, Names of large and small numbers, web page.

InterTran English-Latin Translator, via Stars21.

http://www30.brinkster.com/manfear/office/large.html (Also gives names in German and proposes names based on Greek)

http://g42.org/MiscInfo/numbers.html (Much of the same info as on the previous page)

http://www.unc.edu/%7Erowlett/units/large.html (Much of the same info as on the previous page)

http://mathforum.org/library/drmath/view/59155.html (An introduction for kids)

http://www.grammarstation.com/KnowYourMath/numbers_symbols.html (An introduction for kids)

http://www.ex.ac.uk/cimt/dictunit/notesp.htm (Described the SI prefixes for powers of 1024, but now offline)

http://mathworld.wolfram.com/LargeNumber.html

http://journals.iranscience.net:800/www.newscientist.com/www.newscientist.com/lastword/article.jsp@id=lw77 (comments on uses of zetta- and yotta-)

http://www.lewrockwell.com/orig/kinsella6.html (introduction to the SI units)

http://www.plexos.com/256_bit_CPUs_should_be_enough.htm (xona)

http://jimvb.home.mindspring.com/unitsystem.htm (xona)

http://www.omniglot.com/writing/latin.htm (Latin alphabet)

http://www.freezoneearth.org/Prometheus04/otThree/preot3/bignumbers.htm

Wikipedia, List of numbers (Another version that goes up to 10^{180})
)

### Gödel's undecidable sentence

The discipline of formal logic was described by Aristotle around 350 BC — although it was probably developed well before him. He described the syllogism, an inference of truth based on other established truths. For example:

if all integers are real numbers

and all primes are integers

then all primes are real numbers

Around 1850, George Boole refined formal logic by defining more precise symbols, axioms and rules, the result is called boolean algebra and propositional calculus.

Meanwhile, set theory was being developed by Georg Cantor and others.

Then Friedrich Ludwig Gottlob Frege began developing the predicate calculus. He applied existing formalism techniques to set theory and began developing a complete formalized system of arithmetic, symbolic logic, and set theory from basic axioms and inference rules. This work was just being published as Grundgesetze der Arithmetik vol. 2 when Bertrand Russell (a mathematician who was also well-known as a writer) presented Frege with the following paradox:

Consider the class of all classes that are not a member of itself. Is this class a member of itself or not?

This paradox challenged Frege so completely that he had to withdraw his work because its foundation was flawed.

After toppling Frege's work, Bertrand Russell and Alfred North Whitehead began work on a similar and even more massive project, with corrections to the problems in Frege's approach and using some of the foundation axioms of Giuseppe Peano. Their goal was to make it possible to derive every true theorem in number theory by starting with a set of axioms and a set of inference rules, and methodically applying all the inference rules to the axioms and existing theorems to create new theorems. For examples, see the book Gödel, Escher Bach [41].

The technique of deriving all truth by an automatic process is appealing — it suggests the possibility of automation (e.g. by a mechanical or electronic computer) to eliminate human error in discovering proofs. However, from the start there was much doubt about whether it could ever be used to discover every truth in number theory (if it were possible, the technique would have completeness). For one thing, there was the issue of time and algorithmic complexity: the number of theorems that are provable in such a system is infinite, and all but a very very tiny fraction of those theorems are completely useless (true, but useless).

The goal of formalization was not questioned, and was continued across various fields of mathematics, e.g. by Hilbert.

Gödel finally showed that the Russell/Whitehead approach would not achieve completeness, in fact that no (sufficiently powerful) axiomatic system of number theory can prove all statements which are true in that system. He did this by showing how statements, expressions and theorems in a number theory system S could be used to encode its own symbols, axioms and inference rules. He then demonstrated that there exists a statement G in that system which asserts "Statement G cannot be derived through the axioms and inference rules of formal system S.". If G is indeed inferrable, then it is false — and the formal system S is inconsistent. But if G cannot be inferred, then it is true, and exists as an example of a true statement that cannot be inferred, and therefore demonstrates that S is incomplete. Thus, all sufficiently powerful formal systems of number theory are either inconsistent or incomplete.

It is now generally agreed that this phenomenon of incompleteness is a property of all axiomatic systems.

### Notes on Skewes Number

The Skewes Numbers concern the distribution of primes, specifically how many primes exist that are below a certain number. The Skewes numbers set an upper bound for the point at which the number of primes ends up being less than the approximation "li(x)". See here for a complete description of the problem.

The following table gives the number of primes less than each power of 10:

n | 10^{n} | primes |

1 | 10 | 4 |

2 | 100 | 25 |

3 | 1,000 | 168 |

4 | 10,000 | 1,229 |

5 | 100,000 | 9,592 |

6 | 1,000,000 | 78,498 |

7 | 10,000,000 | 664,579 |

8 | 100,000,000 | 5,761,455 |

9 | 1,000,000,000 | 50,847,534 |

10 | 10,000,000,000 | 455,052,511 |

11 | 100,000,000,000 | 4,118,054,813 |

12 | 1,000,000,000,000 | 37,607,912,018 |

13 | 10,000,000,000,000 | 346,065,536,839 |

14 | 100,000,000,000,000 | 3,204,941,750,802 |

15 | 1,000,000,000,000,000 | 29,844,570,422,669 |

16 | 10,000,000,000,000,000 | 279,238,341,033,925 |

17 | 100,000,000,000,000,000 | 2,623,557,157,654,233 |

18 | 1,000,000,000,000,000,000 | 24,739,954,287,740,860 |

19 | 10,000,000,000,000,000,000 | 234,057,667,276,344,607 |

20 | 100,000,000,000,000,000,000 | 2,220,819,602,560,918,840 |

21 | 1,000,000,000,000,000,000,000 | 21,127,269,486,018,731,928 |

22 | 10,000,000,000,000,000,000,000 | 201,467,286,689,315,906,290 |

(source: Caldwell)

Beyond about n=12 (and slightly larger values as computers get faster) this table cannot be produced by the direct method of finding each prime and counting. There was sufficient interest in this that, as early as 1867, Kulik had completed a table of the smallest factor of every integer (and therefore, giving all the primes) up to slightly above 100,000,000!

Soon after Kulik's table, Meissel found an indirect way to count the exact number of primes up to far higher numbers. His method was simplified by Lehmer in 1959, then improved by Lagarias, Miller and Odlyzko in 1985, and then improved further by Deleglise and Rivat.

The method has been improved to such an extent that it is actually
possible to derive values of pi(x) for really huge values of x
like 10^{100} or 10^{1000}. In 2005 Patrick Demichel found the actual
point at which the pi(x) vs. Li(x) crossover
occurs, 1.397162914×10^{316}. .

### Comparing the Graham-Gardner Number to Chained Arrow Numbers

The following three numbers are listed in ascending order:

3→3→64→2

the Graham-Gardner number

3→3→65→2.

First, note the function-based definition of the Graham-Gardner number:

gn(n) = { hy(3,6,3) for n = 1 { { hy(3, gn(n-1)+2, 3) for n > 1 Graham-Gardner = gn(64)And we will give another similar definition for a function we'll call c3(N):

c3(n) = { 27 for n = 1 { { hy(3, c3(n-1), 3) for n > 1Here is the expansion process for 3→3→N→2:

3→3→1→2 = 27 = c3(1)

3→3→2→2 = 3→3→(3→3→1→2) = 3→3→c3(1) = hy(3,c3(1),3) = c3(2)

3→3→3→2 = 3→3→(3→3→2→2) = 3→3→c3(2) = hy(3,c3(2),3) = c3(3)

and in general,

3→3→N→2 = c3(N)

Now note that:

c3(1) << gn(1) << c3(2)

c3(2) = hy(3, c3(1), 3) << hy(3, gn(1)+2, 3) = gn(2) << c3(2)

and in general

c3(N) = hy(3, c3(N-1), 3) << hy(3, gn(N-1)+2, 3) = gn(N) << c3(N+1)

therefore,

c3(64) = 3→3→64→2 << gn(64) = Graham-Gardner number << 3→3→65→2 = c3(65)

A reader asked me how 3→3→3→3 compares to these:

3→3→3→3

= 3→3→(3→3→2→3)→2

= 3→3→(3→3→(3→3→1→3)→2)→2

= 3→3→(3→3→27→2)→2

= 3→3→c3(27)→2

= c3(c3(27))

therefore 3→3→3→3 is much bigger than the Graham-Gardner Number.

### What Turing and Church Did Not Prove

Turing is often credited with having proven that there are limits to what can be computed by computers, including all conceivable designs of real-world computers. This actually depends a lot on what you mean by computer. Here are some types of computers that a Turing machine cannot emulate:

- Ideal analog computers. This is a computer that manipulates real numbers in the form of analog quantities (like voltages) rather than digits. In an "ideal" analog computer, the quantity being represented is represented exactly, without any error. An example is a circuit that calculates square roots: If you put 2.0 volts on the input wire, it produces 1.4142135... volts on the output.

- Actual analog computers: Any real-world analog computer is subject to various sources of error, including fundamental limits of physics such as the brownian motion of molecules, the quantization of electric charge, and the uncertainty principle (Planck limits).

- Quantum computers: A quantum computer (ideal or actual) exploits properties of quantum mechanics, including most especially the ability for a "particle" such as a photon to be in a state that is indeterminate until that particle interacts with another particle. The conditions of the interaction and the original conditions of the indeterminate state's creation then determine what is observed. It appears that such interactions can be used to perform a sort of massively parallel computation.

- Non-deterministic computers: A nondeterministic computer incorporates some type of phenomenon that is not predictable, in the sense that no amount of advance knowledge can completely predict what the result of a computation will be. Sometimes such phenomena are deliberately exploited, such as a random number generator device based on a Geiger counter (and deriving its randomness from the unpredictable timing of radioactive decay events)

Each of these has one or more properties that make Turing's argument inapplicable. Usually this is some form of unpredictability on a fundamental level. For the ideal analog computers, Turing's argument does not apply because of the cardinality of the reals It is known that Turing planned to create a limits-to-computability proof for such computers, but that he never did (I suspect it was not because such computers are actually without limit!)

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