A recent interchange on Twitter referred to ‘Eternal Goals’.  Instinctively I felt that eternal goals would (by definition) be unachievable and I began to wonder whether there was a rational basis for my gut reaction.  I also began to ponder the qualitative differences between concepts involving ‘eternity’ and those involving the ‘infinite’.  Here are some of my thoughts.

In general the word ‘eternal’ involves ideas about time, whereas the word ‘infinite’ involves ideas about distance, space or quantity. 

We often talk about ‘eternal truth’ by which we mean a persistent truth that is constant and unchanging.  Such a truth is independent of the passage of time; it was true then, is true now, and will be true for ever.  An eternal truth would (arguably) remain true even if there were no people to think about it.

The important thing here is that the concept of ‘eternal’ embraces the present and all things past and future.  It includes our current situation.

The concept of ‘infinite’ is different.  Infinity is so far away in space, or so large in scale, that we cannot possibly relate it to our current situation.  There is a sense in which it is out there, and unattainable.  Unlike the concept of eternity, infinity does not include the here and the now, but is very far away from it indeed.

Infinity has many facets, and some infinities are different from others.  The never ending set of counting numbers (1, 2, 3, 4...) is called ‘countably’ infinite, as is any never ending set of things that can be put into an order and matched exactly one-to-one with counting numbers without any left over.  This is because you can adopt a rule for counting them, even if you have to go on counting for ever without stopping.

Such counting isn’t possible with some infinitely large sets.  For instance there is an infinity of points on a continuous straight line, but however much you tried matching the points on the line with the counting numbers there would always be some (indeed many) points left over.  You can’t even do this sort of matching with all the points on a straight line that is only 1m long.  Because they can't be matched with the counting numbers, the set of points on such a straight line is ‘uncountably’ infinite in size. 

At first sight it feels as if uncountable infinities are, in a sense, bigger than countable infinities.   But how can this be if both are infinitely large?  It’s many years since I studied mathematics and I leave it to better mathematicians than I am to explain whether an uncountably large infinity is larger than a countably large infinity, and why.  They may also like to explain about other kinds of infinity.

As far as I am aware, however, there is only one kind of eternity.  So what of ‘eternal goals’?  I may have wondered off the original point but in working through my thoughts I have concluded that eternal goals are not possible for two reasons:

Firstly, a goal is something that we (or perhaps other sentient beings) strive to achieve.  Unlike ‘truth’ a ‘goal’ must necessarily have a protagonist and cannot exist independently of such a being.  As goals can only persist as long as there are beings to strive towards them, and as species of beings are unlikely to exist eternally, goals themselves are therefore unlikely to be eternal.

Secondly, once goals have been achieved they are no longer goals.  Goals must either be unattainable (in which case they cannot be valid goals or we cease to perceive them as goals), or else they are attainable, in which case they cease to be goals once attained and are therefore not eternal.

What do you think?  I look forward to your comments.