Now offhand you might think that these irre gularities in the progression >m ordinal to ordinal (as these names of infinity are calle d) could be handled by a computer program. That is, there would be a program to produ ce new names in a regular way, and when it ran out of gas, it would invoke the "irregul arity handler", which would supply a new name, and pass control back to the simple one . But this will not work. It turns out that irregularities themselves happen in irregular ways, and one would need o a second-order program-that is, a program which makes new programs which make new names. And even this is not enough. Eventually, a third-order program becomes necessary. And so on, and so on. All of this perhaps ridiculous-seeming co mplexity stems from a deep °theorem, due to Alonzo Church and Ste phen C. Kleene, about the structure of these "infinite ordinals", which says:
There is no recursively related notation-system which gives a name to every constructive ordinal.
What "recursively related notation-systems" are, and what "constructive ordinals" are, we must leave to the more technical sources, su ch as Hartley )gets' book, to explain. But the intuitive idea has been presented. As the ordinals get bigger and bigger, there are irregularities, and irregularities in e irregularitie s, and irregularities in the irregularities in the irregularities, etc. No single scheme, no matter how complex, can name all e ordinals. And from this, it follows that no algorithmic method can tell w to apply the method of Gödel to all possible kinds of formal systems. ad unless one is rather mystically inclined, therefore one must conclude at any human be ing simply will reach the limits of his own ability to 5delize at some point. From ther e on out, formal systems of that complex, though admittedly incomplete for the Gödel reason, will have as much power as that human being.