If you have played with the square root function on an electronic calculator you may have noticed something odd.  It looks like if you take the square root of ANY number that is not actually a perfect square, then you get an irrational number, with the decimal places spilling over the edge of your calculator’s display.  Is this in fact true?  Is the square root of any number that is not itself a perfect square an irrational?  Let’s find out.

We start off in the conventional way by assuming that the square root of 2 IS rational and may be given as m/n where m & n are both integer.

1)  √2 = m/n

2)  m2 = 2 n2

That much we have seen before, but now comes the really clever bit.  Factor m & n into their unique prime factors:

3)  m1 m1 m2 m2 m3 m3 … mr mr = 2 n1 n1 n2 n2 n3 n3 … nr nr

Now here’s the problem.  The prime number 2 will appear an even number of times on the left hand side of the equation (if it appears at all) and an odd number of times on the right.  Since the decomposition into primes is unique the prime number 2 cannot appear an even number of times on one side of the equality and an odd number of times on the other.  So the square root of 2 cannot be written in the form m/n with m & n integer.  The square root of 2 is an irrational number.

A) Although the above was shown to be true for root 2, the same argument of course would hold true for any prime number. The square root of any prime number is an irrational.

B) Replace 2 with any integer that is not a perfect square. Now that integer may be decomposed into its prime factors, and, if the number is not a perfect square, then once again we will have an odd number of primes on one side of the equality and an even number of primes on the other.  The square root of any integer that is not a perfect square is also an irrational.

C) Finally, replace 2 with any perfect square. With even numbers of primes on both sides the solution is trivial.

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Admittedly Cantor was in his writings not very explicit about what he did take the set theoretic universe as a whole to be. One problem is that it is not in every instance clear whether he has a theological or a mathematical conception of absolute infinity in mind. Indeed, he argues that it is the task not of mathematics but of ‘speculative theology’ to investigate what can be humanly known about the absolutely infinite. The following passage, for example, leans heavily to the theological side: I have never assumed a “Genus Supremum” of the actual infinite.  Quite on the contrary I have proved, that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”.  In this quotation, Cantor speaks of the necessity of ‘knowing’ the domain of variation through a ‘definition’. Surely Cantor is merely sloppy here, and we should discount the epistemological overtones. Another slip can be detected in Cantor’s use of the word ‘set’ in this quotation. Cantor means the argument to be applicable not just just to sets but also to absolute infinities. which is by many called “God”. All this is related to the fact that in an Augustinian vein, Cantor takes all the sets to exist as ideas in the mind of God.

And that last sentence rings a bell too :)  For Ramanujan said “An equation for me has no meaning, unless it represents a thought of God”.

We seem to be homing in on something here!

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If God is the Absolute Infinite, the Ein Sof, then I think we are entering very interesting territory.  Why?  Because I don’t believe there is ANYTHING in the physical Universe that is infinite.  I don’t believe there are an infinite number of photons, particles, quarks or neutrinos.  Our Universe it appears is finite in size and contains a finite amount of stuff.  So everything we can know (or ever know) in the real physical world appears to be made out of finite quantities – we won’t find God in the physical Universe.

 

Where do we find Infinities?  The only place I know of where we find Infinity is in mathematics.  Now that’s strange.  We use mathematics to explain the real world to high accuracy, and we even carry out integrations over infinity to give answers that correspond to realities in the real world – and yet infinity does not seem to be part of the real world.

 

So am I saying God is Mathematics?  No I am not.  But can you see that Mathematics might give us a clue as to what God actually is?  The Absolute Infinite was contemplated by Georg Cantor as an infinity that transcended the transfinite numbers.  It should be noted that Cantor equated the Absolute Infinite with God!  Cantor believed that the Absolute Infinite possessed mathematical properties including the reflection principle which states that every property of the Absolute Infinite is also held by some smaller object.  It is sad to relate that Georg Cantor, along with several other famous mathematicians/physicists who dared to venture into the realm of the infinite encountered severe mental problems leading to death.

 

We are now coming to the end of this piece.  How can a finite Mind contemplate and work with infinite quantities?  I suppose the trite answer is that it cannot and it leads to madness, which in itself is extremely interesting, because as we all know “Whom the Gods would destroy, they first make mad”.  But now consider the reason our finite minds can work with the Infinite is due to the reflection principle, so that every property of the Absolute Infinite is also held by some smaller object – ourselves!

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A beautiful clear Moonless sky last night and I put the MegaWASP array onto the Twins – Castor & Pollux.

Castor frame was 20 x 10 minute subs and Pollux frame was 18 x 10 minute subs.

Very pleased with the result :)

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Recently the MegaWASP array grabbed 6 x 2500-second subs on the Rosette nebula.  Noel Carboni added this data together with everything else we had on this object and came up with the result below.  I now consider this one well and truly done.

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Got today’s Earth Science Picture of the Day (EPOD) with a very deep image of the M42 region.  Added narrowband H-alpha and wideband near infrared add a lot to the data in this image.  In particular the infrared data brings out many stars that are not so visible in standard RGB images.

Many thanks to Jim at EPOD who continues to publish my work.

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Even with a blazing Moon, clouds, and everything wanting to go wrong – I managed to get enough data last night for the Caph right hand frame.  So the central star here is Caph in Cassiopeia, a 2-framer using the 200mm lenses and M26C OSC CCDs (this is equivalent to taking an 8-framer with the Sky 90s).  Left hand frame 12 x 10-minute subs, right hand frame 13 x 10-minute subs.  Job done :)

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Clear last night 19/01/2016 but with a blazing Moon, so this didn’t turn out too badly considering.

A 2-frame mosaic using the Canon 200mm lenses and M26C OSC CCDs.  Around 130-minutes of 10-minute subs per frame.

Lots of open clusters, dark lanes, and some nebulosity to be found in this one.

 

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Due to a series of mishaps I only managed to get 15 x 20-minute subs on this one last night using the Sky 90s (should  have got 27 x 20-mins).  This is the IC2087 region in Taurus – and is the biggest region of dark nebulosity I have ever imaged.  Hopefully I can get some more data on this region shortly.

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