Ken Livingstone is a man of contradictions.  Socialist mayor of a London that was at the time going through unprecedented turbo-capitalist growth, he is quite happy to remind us that by 2007 London was overtaking New York in terms of business-friendliness. Here he is at the Cambridge Union Society, surrounded by dinner-jacketed and cocktail-dressed bright young things of the Oxbridge social hierarchy, supping champagne and eagerly awaiting tonight’s debate on class. He is speaking in favour of the motion that “Class still rules Britain”, yet it is rather hard now to see him as the full-throated radical orator of yore. Indeed, he is remarkably soft-spoken throughout the interview, the so called “Red Ken” supping on red wine thoughtfully between his answers. It is true that he has chosen to avoid the traditional evening wear dress code for speakers, opting instead for a sober business suit, but I imagine this is only sensible when one is debating opposite the black tie-clad headmaster of Haberdasher’s Aske’s Boy’s School when they are trying to argue that class isn’t all that important.

He doesn’t claim to have anything much against Oxbridge itself – he’s no Trenton Oldfield.  He never went to university himself, having left school at 16 after failing to get enough O Levels to stay on. But he has employed many Oxbridge graduates in his time, and has found them very talented. His wife is an Oxford graduate.  “If I could make a change, though, I would make it so there isn’t a special admissions process to get in to Oxbridge. I would find a way to test so that anyone who hadn’t had this special tuition and preparation could be on a level pegging.” I suspect he may be thinking of his own generation’s Oxbridge problems rather than ours: he mentions his wife having had to take a year out to prepare for the “Oxford exam”, which of course was harder for poorer children. This sounds more like the process depicted in The History Boys than what we have had to deal with. But at least he is inventive in thinking of alternatives. “We could just have them converted to agricultural colleges” he says, with a twinkle in his eye.

He is quite happy with the term “social mobility”, which in recent times the Labour party seems to be talking about more than “equality”. Doesn’t this mean people have to be downwardly mobile as well as upwardly mobile? “That’s the way of nature, in evolution”, he says, “but in evolution there isn’t this rigged system that holds species back.” He suggests reading Richard Wilkinson and Kate Pickett’s seminal 2009 book The Spirit Level, which shows inequality moving down from a high in 1918 until 1979 and Thatcher, after which it steadily climbs “back to where we were a hundred years ago. And that’s holding a lot of people back”. He doesn’t want anyone, rich or poor, held back from their potential, he says.

He doesn’t see recent stumbles on the Left, such as the RESPECT MP George Galloway’s victory in traditionally Labour Bradford, as a sign of weakness. “He’s had a very strong identity with Muslim voters for decades” he says, remembering when they worked together to get the Labour Party to recognise the Palestinian Liberation Organisation in 1982. He can’t see that it means much outside Bradford. But the recent furore around Galloway’s comments that Julian Assange had been guilty of “bad sexual etiquette” rather than rape, is, he admits “extremely damaging for George and for RESPECT”. Men of a certain age, he says, “don’t express themselves terribly well around issues of rape, and if you’re George Galloway or Ken Clarke the 24-hour media can jump on it.”

What of Ed Milliband's attempt to appropriate the Toryism of the 1950s with his "One Nation" rhetoric?  Despite his adversarial image, he still retains a strange affection for the Tory generation who were in power when he was young. “These people had fought in the war, and had respect for the working class men they had fought with.” It was a Tory party that recognised that all of society had to be carried with them. “And then Thatcher came along and said ‘screw everyone who isn’t one of us.’ And Cameron and Osborne were going through puberty at the time, and they bought all that bullshit.”

Livingstone came into politics from a protesting background, so what can today’s students learn about how to be effective in their struggle against high fees and marketised education? “Protesting has to be peaceful and it has to be imaginative. If you can use humour you might get reported.”  But of course he is sceptical about the likelihood that protests will be covered by a media “owned by four billionaires.” When we have a free press, he says, then you might be able to get through, but while the Barclay brothers, Lord Rothemere and Rupert Murdoch, “four of the most disreputable criminals on the face of the planet” control what we read, the power of protest is limited.

I leave Livingstone to finish his wine and head out into the debating chamber. This short, amiable man is certainly not interested in telling us too loudly about how unjust our privileged existence is, here in the bastion of privilege itself. He’s quite happy to murmur the same familiar bromides about class from the Thatcher days, remaining a cheerful thorn in the side of just about everyone.