The things on their feet are called crampons and they allow you to walk without slipping even on steep icy terrain. For mountaineering they are an essential tool. It’s important to point out that the crevasses climbers cross are on glaciers that form high up in mountains. The giant slab of ice that’s clinging to the mountain is very slowly sliding down it. Therefore the terrain on top of it is an ever changing landscape of giant cracks in the ice that can be deadly. Finding a safe route across this ice does mean having to cross crevasses using equipment that can be carried and tied down and removed just as quickly.
Ladders because they need to be carried up glaciers on peoples backs, their strong and light and long. Spikey shoes (crampons) are used for walking on glaciers and theres not much point in removing them for a ladder crossing, you need them at the lips of a crevasse to stay stable on the ice and you can step on a ladder easily with them, there are no spikes in the middle section generally, so you aim the centre of your boot for each rung while crossing. Also with ropes on each side as handrails and if your tied in its pretty safe, Ice screws or snow anchors used to secure ropes can be very strong.
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