Tony Blair promised in 1999 that everyone would have access to an NHS dentist, an aspiration that now looks further away than ever. In contrast, ministers have made considerable progress with reducing child poverty, which they pledged to abolish, and with increasing NHS spending, which Mr Blair (sitting on a Sunday morning television show sofa) casually said he would bring up to the European Union average, rather like a man promising his wife that she could have a conservatory like the neighbours’.

Mr Blair’s promise took the breath away, as did Gordon Brown’s pledge in his Budget last month to close the funding gap between private and state schools. The only promise of this sort I can remember being fully realised was that by President John F Kennedy to put a man on the moon before the end of the 1960s.

His promise had the advantage that the moon wasn’t going to get further away. The trouble with the other promises is that you may have to run faster just to stay in the same place. For example, child poverty is defined as living in a family that receives less than two-thirds of average income. Unfortunately, the tendency of globalised capitalism is to multiply rewards at the top while depressing those at the bottom. The state therefore has to spend more to fill the gap. Likewise, Mr Blair’s promise to match average EU expenditure on health is hard to keep because other European nations are also increasing their spending.

The Chancellor, being a canny Scot, avoided this trap. He said, “we should agree... that... adjusting for inflation, we raise average investment per pupil to today’s private school level”. In other words, if private school fees rise faster than inflation, Mr Brown (or his successor) can keep his target within reach without busting public finances.

That is not all. The number of school-age children is falling. For private schools, this means a declining market and stiffer competition to attract customers. They fear they will have limited scope over the next decade to continue increasing fees, particularly since they face other pressures.

House prices are rising; middle-class families face growing bills for university fees; and the schools, following an Office of Fair Trading investigation, are now banned from colluding to increase fees. The dread words “increasing teachers’ productivity” - which means increasing class sizes - are now heard at independent school conferences.

In the state sector, falling rolls have different effects. They don’t have to be used, as they were under the Tories in the 1980s, to save money.

Labour can maintain overall spending levels and “invest”, as Mr Brown likes to put it, more in each pupil, thus allowing smaller classes.

So Mr Brown can get a long way towards his target - for which he has not in any case set a date - without lifting a finger. The more you look at it, the more his ambition seems achievable.

I do not, however, think this is necessarily good news for state school teachers. As the funding gap closes, people will inevitably expect the attainment gap to close correspondingly. It won’t. The private sector will continue to select the cleverest, least troublesome children from the most favoured homes and, therefore, stay well ahead. State school teachers will be blamed more than ever for “failure”. Nor will parents, seeing classes as small in the state sector as in the private, necessarily conclude the latter isn’t worth the fees. Small classes are part of the private sector’s attraction, but not the main one. Parents pay so that their children can have bright, well-motivated classmates and be kept away from the rougher sorts.

Mr Brown, like many social democrats, dreams of equality of educational opportunity. It will be achieved, I fear, only when we have greater equality of incomes.