To Infinity and Beyond?

“Lord, You have been our dwelling place in all generations. Before the mountains were brought forth, or ever You had formed the earth and the world, even from everlasting to everlasting, You are God.” – Psalm 90: 1, 2

“…Your years are throughout all generations. Of old You laid the foundation of the earth, and the heavens are the work of Your hands. They will perish, but You will endure; yes, they will all grow old like a garment; like a cloak You will change them, and they will be changed. But You are the same, and Your years will have no end.
– Psalm 102: 24 – 27

1.  The Bible presents us with a God Who has always existed and will always exist – He is infinite!

  • He is called the One without end

“The Eternal God is your refuge…” – Deuteronomy 33: 27

“But the Lord is the true God; He is the living God and the everlasting King…” – Jeremiah 10: 10

“Now to the King eternal, immortal, invisible, to God who alone is wise, be honor and glory forever and ever. Amen.”  – I Timothy 1: 17

2.  Terms such as eternal, everlasting, immortal, invisible are all ascribed to Him

  • These are all concepts in keeping with infinity – all mathematical traits by their nature

“God, Who made the world and everything in it, since He is Lord of heaven and earth, does not dwell in temples made with hands.”  – Acts 17: 24

“Am I a God near at hand,” says the Lord, “And not a God afar off? Can anyone hide himself in secret places, So I shall not see him?” says the Lord; “Do I not fill heaven and earth?” says the Lord.” – Jeremiah 23: 23, 24

“But will God indeed dwell on the earth? Behold, heaven and the heaven of heavens cannot contain You. How much less this temple which I have built.” – I Kings 8: 27

3.  Not only is He the only Being Who has always existed, but He is everywhere present!

  • Omnipresence, omnipotence, and omniscience are all principles and traits of infinity
  • God alone is infinite

“Great is our Lord, and mighty in power; His understanding is infinite.”  – Psalm 147: 5


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4.  In this text, God’s understanding or wisdom is connected directly with this concept of infinity

  • It is uncertain as to who penned the words of this Psalm 147
  • There are three likely candidates however: David (@1000BC), Haggai (@520BC), or Zechariah (@520BC)
  • The word used for “infinite” is the Hebrew transliteration “micpar”, meaning “without number, innumerable, never ending, without measure”

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5.  Regardless of who the author may be, we do know that the Scripture discusses this essentially mathematical concept of infinity long before it was presented in secular or non-biblical circles

  • That honor goes to Aristotle (384 – 322 BC) who is the first documented person outside of the Bible to conceptualize infinity

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6.  Still, proving that such a concept could actually exist would be left undone for more than 2,000 years

  • In the late 1800’s, mathematicians began to tackle theorems, axioms, and proofs having to do with infinity
  • In 1874, Russian born / German raised mathematician George Cantor presented a paper, On a Characteristic Property of All Real Algebraic Numbers

√ Source: Philip Johnson, A History of Set Theory (1972)

  • It was this paper that laid out the premise for “Set Theory” – which is the basis for an entire field, called “Pure Mathematics”

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7.  Cantor’s paper put forth a premise that there is a basic type of infinity – namely, an unending list of Natural Numbers: 1,2,3,4,5,6,7…

  • This he referred to as a “Countable Infinity”

8.  Cantor also laid out that any other infinity set you can think of would also be countable to infinity by putting its members in a 1:1 correspondence with the natural numbers

  • For example, if your set was to include only even numbers: 2,4,6,8,10,12,14 … your set could still go on forever and this too would be an unending list countable to infinity
  • The same could be done if your set was to include only odd numbers: 1,3,5,7,9,11,13 …
  • Each of these sets as well would be a “Countable Infinity”

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9.  Mathematicians began to realize the mind bending consequences of this however

  • Your intuition tells you that the set of even numbers (namely 2,4,6,8,10,12,14) should only possess half of the numbers within the set of your natural numbers (namely 1,2,3,4,5,6,7…)
  • Yet, since both lists can go on forever, this is no longer the case – both Sets are INFINITE
  • This is what is classically referred to as an “axiom” (a mathematical statement that is accepted as being true due to its self-evidence, even though it can not be proven)
  • Galileo, a few centuries earlier, was probably the first to see the paradox in all of this (i.e., it seemed to defy logic)

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10.  Cantor reconciled the paradox by showing that as long as you can create a 1:1 ratio between a particular infinite set (e.g., set of endless odd numbers) and the set of natural numbers, then each set must be countable to infinity

  • 1 in the odd set would correspond to 1 in the natural set
  • 3 in the odd set would correspond to 2 in the natural set
  • 5 in the odd set would correspond to 3 in the natural set
  • And so on and so forth…

11.  This same logic applies to Rational Numbers, that is fractions

  • As long as you can count them or create a set based on identified logic, these too will go on to infinity
  • For example, if your set was to include all fractions where the numerator and denominator equal the next number in the natural number series – it would look as follows:
    • First write down all the fractions for which this sum is 2 (there is only one, 1/1), then all the ones for which it is 3 (1/2 and 2/1), and so on
    • Each time you are counting only a finite number of fractions (the number of fractions p/q where p+q=n is n-1)
  • This is a fool-proof formula for counting all rational numbers: you won’t miss any

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12.  Similar to our experience with Natural Numbers, your intuition would lead you to believe that there are a lot more Rational Numbers (fractions) than there should be Natural Numbers – but this also is incorrect

  • Both lists can go on forever and are therefore INFINITE


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13.  Cantor took his study even further and showed that there is also an “Uncountable Infinity”

  • For example, if your set were to include within a set all Real Numbers – that is, all Natural Numbers (like, 1,2,3,4,5, etc…); all Rational Numbers (such as fractions or decimals); all Irrational Numbers (numbers that can not be written as a simple fraction, such as “Pi” or the square root of 3); this too would go on to infinity
  • However, since it is not countable based on some formula, it is considered an “Uncountable Infinity” or a “Continuum”

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14.  As a result, Cantor concluded that you could go and on, forever finding bigger and bigger sets always leading to Infinity

  • “This never-ending tower of infinities pointed towards something called absolute infinity — an unreachable summit of the tower of infinities.”
  • Cantor’s paper concluded that not only would such a summit of infinities be potential, but it must exist in actuality

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15.  In 1908, a German mathematician by the name of Ernst Zermelo proposed the first Axiomatic Set Theory based on Cantor’s work

  • This consisted of 7 Axioms or self-evident truths – one of which speaks to the existence of an Infinity Set
  • In 1922, a German Jew by the name of Abraham Fraenkel contributed to Zermelo’s Set Theory so that it could better address shortcomings related to Ordinal Numbers

16.  Today, we refer to their collective work as the Zermelo Fraenkel Set Theory

  • Its “Axiom of Infinity” is a mathematical demonstration that “Infinity” must exist in actuality
  • Remember – an Axiom is a mathematical statement that is accepted as being true due to its self-evidence, even though it can not be proven
  • It would seem though that shades of Psalm 19 and Romans 1: 19 – 20 are bleeding through into mathematics

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17.  The Bible came to this conclusion a long time ago – it refers to the Infinite as the Lord God